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THE JOURNAL OF FINANCE • VOL. LXI, NO. 6 • DECEMBER 2006
Estimating the Gains from Trade
in Limit-Order Markets
a
BURTON HOLLIFIELD, ROBERT A. MILLER, PATRIK SANDAS,
and JOSHUA SLIVE∗
ABSTRACT
We present a method to estimate the gains from trade in limit-order markets and provide empirical evidence that the limit-order market is a good market design. Using
observations on order submissions and execution and cancellation histories, we estimate both the distribution of traders’ unobserved valuations for the stock and latent
trader arrival rates. We use the resulting estimates to compute the current gains from
trade, the gains from trade in a perfectly liquid market, and the gains from trade with
a monopoly liquidity supplier. The current gains are 90% of the maximum gains and
150% of the monopolist gains.
THE MAJORITY OF THE WORLD’S STOCK EXCHANGES operate some form of a limit-order
market. A feature of good market design is that traders realize most of the
potential gains from trade. We develop a method for identifying and estimating
the gains from trade in a limit-order market and apply our method to a sample
from a particular limit-order market, the Vancouver Stock Exchange (VSE).
We find that gains from trade in the VSE are approximately 90% of the gains
from trade in a perfectly liquid market and approximately 50% more than gains
from trade in a market in which liquidity is supplied by a profit-maximizing
monopolist. Our results therefore provide new empirical evidence that the limitorder market is a good market design.
A large number of experimental studies document that the gains from trade
in a double auction are close to the maximum gains from trade. See, for example, Cason and Friedman (1996) or the survey by Holt (1995). Similarly, our
empirical results show that the limit-order market—a market design similar to
∗ Hollifield and Miller are from Carnegie Mellon University, Sandaa s is from the University of
Virginia and Centre for Economic Policy Research, and Slive is from HEC Montréal and Center
a
for Research on e-finance. Sanda s did part of this research as a faculty member at the University
of Pennsylvania and as a visiting economist at the New York Stock Exchange. The comments and
opinions expressed in this paper are the authors’ and do not necessarily reflect those of the directors,
members, or officers of the New York Stock Exchange. We thank the Carnegie Bosch Institute at
Carnegie Mellon University, Institut de Finance Mathématique de Montréal, the Rodney L. White
Center for Financial Research at Wharton and the Social Science and Humanities Research Council
of Canada for providing financial support, and the Vancouver Stock Exchange for providing the
data. Comments from participants at many seminars and conferences, an anonymous referee, an
associate editor, the editor (Rob Stambaugh), Cevdet Aydemir, Dan Bernhardt, Giovanni Cespa,
Pierre Collin-Dufresne, Larry Glosten, Bernd Hanke, Ohad Kadan, Pam Moulton, Jason Wei, and
Pradeep Yadav have been helpful to us.
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The Journal of Finance
that of the double auction—is remarkably efficient. To our knowledge, our empirical estimates of the gains from trade are the first such estimates computed
using field data from a limit-order market.
Our sample comes from the audit tapes of the VSE and contains traders’
order submissions and execution histories. The VSE is organized as a limit order
market with a market design very similar to that of the London Stock Exchange,
Euronext, the Stockholm Stock Exchange, the Toronto Stock Exchange, and the
Archipelago Exchange. The VSE is primarily a venture-capital exchange and
around two-thirds of its stock listings are in the natural resource industry. It is
similar in its objectives to AIM in London, the Second Market and New Market
components of Euronext, Nya Marknaden in Stockholm, and the Tokyo Stock
Exchange’s Mother system (High-Growth and Emerging Stock Markets). All of
these markets focus on emerging companies that wish to have access to public
equity financing but that do not meet the listing requirements of more senior
exchanges.
The companies trading on these venture exchanges have characteristics that
differ from a typical New York Stock Exchange stock. They tend to be smaller
and less frequently traded, for example. However, it is just these types of differences that make the VSE a particularly appropriate context within which to
examine the gains from trade. Investors will care most about the performance
of the trading system where liquidity is low and highly variable. In turn, stocks
with highly variable liquidity will have large variation in the gains from trade.
Studying such stocks gives us a good chance of being able to identify and explain
variations in the gains from trade.
Our model is an extension of the model of traders’ optimal order submission
a
in limit-order markets in Hollifield, Miller, and Sanda s (2004). In that model,
the traders’ order submissions depend on their valuations for the stock and the
trade-offs across execution probabilities, picking-off risks, and order prices for
alternative order submissions. Hollifield et al. (2004) derive and test restrictions implied by the model using a semiparametric test and their empirical
evidence lends some support to the model framework. Here, we extend their
model to continuous time and include an order-execution cost. The extension to
continuous time allows our model to deal with the selection problem that arises
when some traders find it optimal not to submit an order, and including a more
f lexible cost structure allows our model to deal with the rejections reported in
Hollifield et al. (2004).
The model we use captures a key feature of limit-order markets, namely,
the ability of traders to choose whether to submit market or limit orders. In
that sense, our model is consistent with the theoretical models of traders’ order
submissions in limit-order markets of Foucault (1999), Foucault, Kadan, and
Kandel (2005), Parlour (1998), and Wald and Horrigan (2005). However, the
additional f lexibility of our model makes it suitable for empirical work. For
example, the traders in our model may choose from multiple limit-order prices,
unlike the traders in Parlour (1998); the traders’ limit orders in our model may
last for multiple periods, unlike the limit orders in Foucault (1999); and limit
Estimating the Gains from Trade
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orders in our model may be canceled, unlike the limit orders in Foucault et al.
(2005).
A number of researchers have also developed theoretical models abstracting from the choice between market and limit orders. Glosten (1994) shows
that in a competitive environment the limit-order market provides enough liqa
uidity to discourage entry by other competing market designs. Sanda s (2001)
tests and rejects the restrictions implied by one version of the Glosten (1994)
model with discrete prices and a time priority rule as in Seppi (1997). Biais,
Martimort, and Rochet (2000) study imperfect competition among a finite number of traders that submit limit-order schedules and show that the limit-order
book of Glosten (1994) obtains when the number of limit-order submitters becomes large. Glosten (2006) relaxes the assumption of perfect competition, and
shows that under imperfect competition, the limit order market is an optimal
market design considering the gains from trade of both traders who submit
market orders and traders who submit limit orders.
Our model differs from Biais et al. (2000), Glosten (1994, 2006), and Seppi
(1997) in two respects. First, we allow traders to choose between market and
limit orders, and second, we model the dynamics of individual order submissions. We do not allow for endogenous market-order quantities or asymmetric
information, but we do allow limit orders to face picking-off risk. As in Glosten
(2006), we consider the gains from trade that accrue to both traders who submit
market orders and traders who submit limit orders. Our empirical evidence on
the efficiency of the limit-order market complements Glosten’s (2006) theoretical results on the efficiency of the limit-order market.
Many studies document that the empirical frequency of limit- and marketorder submissions changes with market conditions. Using a sample from the
Paris Bourse, Biais, Hillion, and Spatt (1995) show that traders are more likely
to submit limit orders in markets with wide spreads or thin limit-order books.
Griffiths et al. (2000) report similar findings for the Toronto Stock Exchange,
and Ranaldo (2004) reports similar findings for the Swiss Stock Exchange. Using a sample from the New York Stock Exchange, Harris and Hasbrouck (1996)
show that traders are more likely to submit limit orders when the expected payoffs from submitting limit orders increase. We extend the literature by using
the empirical variation in the frequency of limit and market order submissions
to empirically link traders’ order submissions to their valuations. Without an
empirical link between traders’ valuations and their order submissions, it is
not possible to estimate the gains from trade.
Our sample contains both the traders’ order submissions and the execution and cancellation histories of the traders’ order submissions. In our model,
traders’ gains depend on their valuations for the stock. We apply a discrete
choice model that links the traders’ valuations to the traders’ observable order
submissions using the expected payoffs from the different order submissions
that the traders could have made. We use the discrete choice model to estimate
the distribution of the traders’ valuations, the expected payoffs from alternative order submissions, and the traders’ optimal order submission strategy. We
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2756
use the resulting estimates to compute the gains from trade in the limit-order
market, the maximum gains from trade, and the monopoly gains from trade.
The maximum gains from trade are the gains from trade that the traders
would obtain in a perfectly liquid market, and the monopoly gains from trade
are the gains from trade that the traders would obtain if liquidity were supplied by a profit-maximizing monopolist. The maximum gains from trade provide an upper bound on the gains from trade in any mechanism and a natural
benchmark against which to measure the efficiency of a market’s design. In our
sample of stocks from a relatively illiquid exchange, the gains from trade in the
limit-order market are approximately 90% of the maximum gains from trade
and approximately 150% of the monopoly gains from trade. In this respect, the
limit-order market is a good market design—traders in the Vancouver Stock
Exchange realize most of the potential gains from trade.
Section I describes the theoretical model and Section II describes the current,
maximum, and monopolist gains from trade in the theoretical model. Section III
describes our sample and reports estimates of the model parameters. Section IV
reports our estimates of the current, maximum, and monopolist gains from
trade and Section V concludes. The appendices contain the proofs and additional
technical material.
I. Model
Our model captures several key features of trading in a limit-order market.
Any trader can submit market and limit orders, and all traders face the same
order submission and order execution rules—there are no designated market
makers or other traders with special quoting obligations or trading privileges.
Moreover, the market is transparent—all traders observe the limit-order book
and general market conditions when making their order submission decisions.
Finally, all trades involve a limit order being executed by a market order, with
limit orders executed according to strict price and time priority.
A. Model Structure
The model is set in continuous time. Traders arrive sequentially and differ
in their valuations of the stock. The finite-dimensional vector xt denotes the
exogenous state variables that determine the conditional trader arrival rate
and the conditional distribution of the traders’ valuations. The exogenous state
variables follow a stationary Markov process.
The probability that a trader arrives between t and t + dt is given by
Pr (Trader arrives in [t, t + dt) | xt ) = λ(t; xt ) dt,
(1)
which can be interpreted as
lim
t↓0
Pr (Trader arrives in [t, t + t) | xt )
= λ(t; xt ).
t
(2)
Each trader is risk neutral with valuation of the stock of vt . We decompose vt
as follows:
vt = y t + ut .
(3)
Estimating the Gains from Trade
2757
The random variable yt is the common value of the stock at t; yt may be interpreted as the traders’ common time t expectation of the liquidation value of the
stock. Innovations in the common value are drawn from a stationary process,
with possibly time-varying conditional moments.
The random variable ut is a trader’s private value for the stock. Different
traders have different private values, creating the opportunity for gains from
trade. The private value is an independent random variable drawn from the
conditional distribution
Pr (ut ≤ u | xt ) ≡ G(u | xt ),
(4)
with density g(· | xt ). Once a trader arrives at the market, his private value
remains fixed while he has an order outstanding.
The conditional trader arrival rates, the conditional distributions of the innovations in the common value, and the conditional distributions of the private
values all depend on the exogenous state variables. The exogenous state variables therefore determine the intensity of trader arrivals, the distribution of
changes in the stock’s common value, and the aggregate willingness of traders
to pay a price that deviates from the common value in order to obtain immediate order execution. An example of such a state variable that we have in mind
is lagged common value volatility. For example, following a period of high common value volatility the intensity of trader arrival, the future common value
volatility, and the traders’ aggregate willingness to pay a price that deviates
from the common value for immediate order execution may all change.
A trader who arrives at t has a single opportunity to submit either a market
order or a limit order for q shares. We normalize q to one unit. While we can
allow the order quantities to vary exogenously across traders in our model, we
present the case of unit quantity to reduce notation. In our empirical work, we
condition on the observed order quantities. By assuming that each trader has
only one order submission opportunity, we abstract from a trader’s endogenous
future cancellation and resubmission decisions.
The trader’s order submission at t depends on the common value, yt , his
private value, ut , and his information. The trader’s information is captured by
the exogenous state variables, xt , and a finite-dimensional vector of endogenous
state variables, wt . Let zt ≡ (xt , wt ) denote the state vector that represents the
trader’s information. The state vector zt follows a stationary Markov process.
The exogenous state variables xt predict the future trader arrival rates, the
distribution of innovations in the common value, and the distribution of future
traders’ valuations; the exogenous state variables may therefore predict the
execution probabilities and picking-off risks (i.e., the expected loss from such
executions) of a new order submission.
The endogenous state vector wt includes information about the current limitorder book and past order submission activity. For example, the current bid-ask
spread is likely to predict the execution probability for limit orders and so the
bid-ask spread is an element of wt . Similarly, a limit order submitted with few
limit orders in the book is likely to have a different execution probability and
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picking-off risk than a limit order submitted with many limit orders in the book.
The endogenous state vector wt therefore includes the current limit-order book.
The endogenous state vector wt also includes information that is useful in
predicting the distribution of cancellations for the orders in the book. Limit
orders are executed according to price and time priority. As a consequence, the
execution probability of a newly submitted limit order depends on the cancellation probabilities of the existing limit orders in the book. Suppose that the
conditional probability that a limit order is canceled depends on how long the
limit order has been in the book. In this case, the average age of limit orders in
the book helps predict the probability that current limit orders in the book are
canceled in the future. For example, past order submission activity is correlated
with the age of the unexecuted orders in the book, and so past order submission
activity is useful to a trader in predicting the execution probabilities of a new
limit order submission.
buy
The decision indicators dsell
t,s ∈ {0, 1} for s = 0, 1, . . . , S and dt,b ∈ {0, 1} for
b = 0, 1, . . . , B denote the trader’s order submission at t, where s and b index
the finite set of available order submissions: S < ∞ and B < ∞. Let psell
t,s denote
buy
sell
the sell price associated with dt,s , and let pt,b denote the buy price associated
buy
with dt,b . If the trader submits a sell market order, then the order price is the
sell
best bid quote, psell
t,0 , and dt,0 = 1. If the trader submits a sell limit order at
sell
the price psell
t,s , s ticks above the current best bid quote, then dt,s = 1. Similar
definitions apply to the buy side. If the trader does not submit any order at
buy
time t, then dsell
t,s = 0 for all s and dt,b = 0 for all b.
A limit order is either executed or canceled. We define two latent random
times for each order: The latent cancellation time, t + τcancel , and the latent
execution time, t + τ execute . The order is executed at t + τ execute if τexecute ≤ τcancel
and the order is canceled at t + τ cancel if τexecute > τcancel . Orders do not last longer
than T < ∞, thus the random variable τ cancel is bounded from above by T <
∞. The distributions of latent times describe the uncertainty about the limit
order’s outcome.
There is an order submission cost of co ≥ 0 for all types of order submissions.
There is also an order execution cost of ce ≥ 0: The trader pays a cost of ce
when the order executes. The costs, co and ce , do not depend on the trader’s
valuation, nor on the trader’s order submission at t. One interpretation of ce
is that it represents the commission on the trade. With ce = 0 the payoff from
order submissions at t are the same as in Hollifield et al. (2004).
Suppose that a trader with valuation vt = yt + ut submits a buy limit order
buy
buy
b ticks below the ask quote at price pt,b : dt,b = 1. The conditional distribution
of the latent cancellation time depends on the state vector, zt , and on the order
submission itself, but it does not depend on the trader’s private value. Conditional on zt , the latent cancellation time is independent of all other random
variables in the model. One interpretation of the conditional independence assumption is that traders find it too costly to continuously monitor their limit
orders. The probability distribution of the latent cancellation time is
buy
buy
Pr t + τcancel ≤ t + τ z t , d t,b = 1 = Fcancel τ z t , d t,b = 1 .
(5)
Estimating the Gains from Trade
2759
The conditional distribution of the latent execution time depends on the state
buy
vector, zt , and on the order submission, dt,b = 1, but not on the trader’s private
value. The probability distribution of the latent execution time is
buy
buy
Pr t + τexecute ≤ t + τ z t , d t,b = 1 = Fexecute τ z t , d t,b = 1 .
(6)
The execution time depends on the trader’s order submission, future order cancellations, and the arrival of future traders and their order submissions. The
execution time therefore depends on how future traders behave given their valuations and the order books and information they face, and the latent execution
times depend on future traders’ order submissions.
The execution time distribution is defined conditional on the current trader’s
order submission. The conditional distribution of the execution time is therefore different for different order submissions that the trader could make. By
conditioning on potential order submissions, the trader considers the feedback
from his current order submission to future traders’ order submission decisions.
Define the indicator function for order execution as
1, if t + τexecute ≤ t + τcancel ,
It (τexecute ≤ τcancel ) =
(7)
0, otherwise.
buy
The realized utility from submitting a buy order at price pt,b is
buy
It (τexecute ≤ τcancel ) y t+τexecute + ut − pt,b − ce − co
buy
= It (τexecute ≤ τcancel ) y t + ut − pt,b − ce
+ It (τexecute ≤ τcancel )( y t+τexecute − y t ) − co .
(8)
The first term on the first line is the indicator for execution multiplied by the
payoff at execution and the second term is the order submission cost.
Define
buy
buy
ψb (z t ) ≡ E It (τexecute ≤ τcancel ) z t , d t,b = 1
(9)
as the execution probability for the order. For a market order, the execution
probability is one.
An order may execute after there has been a change in the stock’s common
value; we refer the expected loss from such executions as the picking-off risk.
Define
buy
buy
ξb (z t ) ≡ E It (τexecute ≤ τcancel )( y t+τexecute − y t ) z t , d t,b = 1
(10)
as the picking-off risk for the order. Since a market order executes immediately, the picking-off risk for a market order is zero. Using the law of iterated
expectations, the picking-off risk simplifies to
buy
buy
buy
ξb (z t ) = E ( y t+τexecute − y t ) It (τexecute ≤ τcancel ) = 1, z t , d t,b = 1 ψb (z t ). (11)
The picking-off risk is the expected change in the common value between the
time of the order submission and the time of the order execution, conditional
on execution, multiplied by the probability that the order executes.
The Journal of Finance
2760
The conditional distribution of the latent cancellation times, the conditional
distribution of the latent execution times, and the expected change in the common value conditional on execution all depend on the state vector, zt . As a
consequence, the execution probabilities and picking-off risks also depend on
the state vector.
buy
The trader’s expected utility from submitting a buy order at price pt,b is the
expected value of equation (8), conditional on the trader’s information, which,
using the definitions of the execution probability and picking-off risk, is
buy
buy
buy
buy
Ub ( y t + ut ; z t ) = ψb (z t ) y t + ut − pt,b − ce + ξb (z t ) − co .
(12)
Similarly, the expected utility of submitting a sell order at psell
t,s is
sell
Ussell ( y t + ut ; z t ) = ψssell (z t ) pt,s
− y t − ut − ce − ξssell (z t ) − co .
(13)
The trader’s order submission strategy maximizes his expected utility,
max
buy
S
sell
{d t,s
},{d t,b } s=0
sell sell
d t,s
Us ( y t + ut ; z t ) +
B
buy
buy
d t,b Ub ( y t + ut ; z t ),
(14)
b=0
subject to
S
s=0
sell
d t,s
+
B
buy
d t,b ≤ 1.
(15)
b=0
Equation (15) is the constraint that at most one submission is made at t.
B. Optimal Order Submission Strategies
buy∗
Let {dsell∗
(yt + ut ; zt ), db (yt + ut ; zt )} be the optimal order submission strats
egy, where the trader’s optimal order submission is a function of the trader’s
valuation and the state vector zt .
Hollifield et al. (2004) show that the optimal order submission strategy has a
monotonicity property. Traders with high private values submit buy orders with
high execution probabilities. Traders with low private values submit sell orders
with high execution probabilities. Traders with intermediate private values
either submit no order, or submit buy or sell limit orders with low execution
probabilities.
The optimal order submission strategy is represented in terms of threshold
valuations. We can partition the set of valuations into intervals. All traders
whose valuations lie within the same interval submit the same order. In order to
characterize the intervals, we define a set of threshold valuations that mark the
boundaries of the intervals. We determine a trader’s optimal order submission
simply by identifying which interval the trader’s valuation is in.
buy
Define the threshold valuation θb,b (zt ) as the valuation of a trader who is
buy
indifferent between submitting a buy order at price pt,b and a buy order at
buy
price pt,b , that is,
Estimating the Gains from Trade
buy
θb,b (z t )
=
buy
pt,b
+ ce +
buy
buy
buy buy
buy
pt,b − pt,b ψb (z t ) + ξb (z t ) − ξb (z t )
buy
buy
ψb (z t ) − ψb (z t )
2761
.
(16)
Similarly, the threshold valuation for indifference between a buy order at price
buy
pt,b and not submitting an order is
buy
buy
buy
θb,no (z t ) = pt,b + ce −
ξb (z t ) − co
buy
ψb (z t )
,
(17)
the threshold valuation for indifference between a sell order at price psell
t,s and a
sell
sell order at price pt,s is
sell
sell
sell
(z t )
ψs (z t ) + ξssell (z t ) − ξssell
pt,s − pt,s
sell
sell
θs,s (z t ) = pt,s − ce −
,
(18)
ψssell (z t ) − ψssell
(z t )
the threshold valuation for indifference between a sell order at price psell
t,s and
not submitting any order is
sell
sell
θs,no
(z t ) = pt,s
− ce −
ξssell (z t ) + co
,
ψssell (z t )
(19)
and the threshold valuation for indifference between a sell order at price psell
t,s
buy
and a buy order at price pt,b is
θs,b(z t )
buy buy
buy
buy
sell sell
pt,b ψb (z t ) + pt,s
ψs (z t ) + ce ψb (z t ) − ψssell (z t ) − ξb (z t ) + ξssell (z t )
=
.
buy
ψssell (z t ) + ψb (z t )
(20)
It may be the case that some order submissions are not optimal for any trader.
Let S ∗ (z t ) = {s0 (z t ), s1 (z t ), s2 (z t ), . . . , sS (z t )} index the set of sell orders that are
optimal for some trader at state zt sorted by their execution probabilities: 1 ≥
(z t ) > ψssell
(z t ) > · · · > ψssell
(z t ). Define a sell limit order submitted at
ψssell
0 (z t )
1 (z t )
S (z t )
sell
price pt,sS (z t ) as the marginal sell order. We assume that a sell market order is
optimal for traders with some private values and that some sell limit order
is optimal for traders with different private values; S ∗ (z t ) has at least two
elements. Similarly, let B ∗ (z t ) index the set of buy orders that are optimal for
some trader in state zt , also sorted by execution probabilities, and define a buy
buy
limit order submitted at pt,bB (z t ) as the marginal buy order.
bu y
A trader with a valuation lower than θbB (z t ),no (z t ), the threshold between a
marginal buy order and no order submission, receives a lower expected payoff
from submitting any buy order than from submitting no order. A trader with a
valuation greater than θssell
(z t ), the threshold between a marginal sell order
S (z t ),no
and no order submission, receives a lower expected payoff from submitting any
The Journal of Finance
2762
bu y
sell order than from submitting no order. If θssell
(z t ) ≤ θbB (z t ),no (z t ), then a
S (z t ),no
bu y
trader with a valuation between θssell
(z t ) and θbB (z t ),no (z t ) submits no order.
S (z t ),no
buy
θbB (z t ),no (z t )
buy
θbB (z t ),no (z t )
If
≤ θssell
(z t ), then
≤ θsS (z t ),bB (z t ) (z t ) ≤ θssell
(z t ), and
S (z t ),no
S (z t ),no
a trader with any possible valuation submits some order. We therefore define
the marginal thresholds for sellers and buyers as
buy
buy
θmarginal (z t ) = max θsS (z t ),bB (z t ) (z t ), θbB (z t ),no (z t ) ,
(21)
sell
θmarginal
(z t ) = min θsS (z t ),bB (z t ) (z t ), θssell
(z
)
.
t
S (z t ),no
Using the definition of the thresholds, the sell side of the optimal order submission strategy is
for s ∈
/ S ∗ (z t ),
d ssell∗ ( y t + ut ; z t ) = 0,
d 0sell∗ ( y t
d ssell∗
( yt
i (z t )
+ ut ; z t ) =
+ ut ; z t ) =
(z t ),
if − ∞ ≤ y t + ut < θssell
0 (z t ),s1 (z t )
0,
otherwise,
⎧
⎪
⎨ 1,
⎪
⎩
d ssell∗
( y t + ut ; z t ) =
S (z t )
1,
(22)
(23)
/ {0, sS (z t )} and
if si (z t ) ∈
θssell
(z t ) ≤ y t + ut < θssell
(z t ),
i−1 (z t ),si (z t )
i (z t ),si+1 (z t )
0,
otherwise,
1,
sell
(z t ) ≤ y t + ut < θmarginal
(z t ),
if θssell
,
S−1 (z t ) sS (z t )
0,
otherwise,
(24)
(25)
with the buy side defined similarly.
II. The Gains from Trade
Each trade involves either a sell limit order executing with a buy market order
or a sell market order executing with a buy limit order. The gains from a trade
are the sum of the traders’ realized utilities from the trade. Using equation (8),
the gains from trade for a trade at t + τ between a sell market order submitted
at t + τ by a trader with valuation usell
t+τ and a buy limit order submitted at t by
buy
a trader with valuation ut are
sell
buy
buy
pt+τ,s − y t+τ − usell
− pt,b − ce − co
t+τ − ce − co + y t+τ + ut
buy
= −usell
− ce − co .
(26)
t+τ − ce − co + ut
buy
The second line follows because psell
t+τ,s = pt,b , since the sell market order executes with the buy limit order. The gains from trade for a trade between a sell
limit order and a buy market order are computed similarly.
The gains from trade depend neither on the price nor on the common value,
since these are simply transferred between the traders. Rather, the gains from
trade depend on the difference between the traders’ valuations at the time of
Estimating the Gains from Trade
2763
buy
the trade. The buyer’s contribution to the gains from trade is ut − ce − co and
the seller’s contribution to the gains from trade is −usell
t+τ − ce − co . If a trader
submits an order that does not execute, he reduces the gains from trade by co .
In the example above, we consider the gains from trade for one possible outcome for the buy limit order submitted by the trader at t. It is also useful to
consider the ex ante gains from trade in a given state before the trader’s valuation is drawn. Using the distribution of the traders’ valuations for the stock
and the optimal order submission strategy, we compute expectations over the
traders’ valuations, the optimal order submissions, and the outcomes of their
order submissions. We define the current gains from trade as the expected
contribution to the gains from trade in state zt . Using the execution probabilities, the traders’ optimal order submission strategy, and the distribution of
the traders’ valuations, the expected contribution to the gains from trade for a
trader arriving at state zt is
⎤
⎡
S
sell
sell∗
d s ( y t + ut ; z t ) ψs (z t )(−ut − ce ) − co
⎥
⎢
⎥
⎢ s=0
⎥
⎢
Current gains (z t ) = E ⎢
zt ⎥ .
⎢
B
⎥
buy
⎦
⎣
buy∗
+
d b ( y t + ut ; z t ) ψb (z t )(ut − ce ) − co b=0
(27)
The current common value, the limit and market order prices, and the state
vector zt enter through their effects on the traders’ optimal order submission
strategy.
The maximum gains from trade are determined by finding the post-trade
allocation of the stock among the traders that results in the maximum expected
gains from trade. The maximum gains from trade may not be achievable by
any mechanism because of the inherent frictions caused by traders arriving
sequentially with trading opportunities that last for a finite period of time.
Incentive compatibility issues will typically further reduce the gains from trade
attainable in any feasible mechanism. The maximum gains from trade is easy
to compute and provides a useful upper bound on the gains from trade in any
feasible mechanism.
To describe a stock allocation, define the sell indicator function as
1, if a trader with private value ut sells the stock in state xt ,
sell
I (ut ; xt ) =
0, otherwise,
(28)
and define the buy indicator function Ibuy (ut ; xt ) similarly.
Using the sell and buy indicators, the allocation that maximizes the gains
from trade solves
max
{I sell (ut ;xt ), I buy (ut ;xt )}
E I sell (ut ; xt )(−ut − ce − co ) + I buy (ut ; xt )(ut − ce − co ) xt , (29)
The Journal of Finance
2764
subject to
I sell (ut ; xt ) + I buy (ut ; xt ) ≤ 1, for all ut ,
E I sell (ut ; xt ) xt = E I buy (ut ; xt ) xt .
(30)
(31)
Equation (30) is the constraint that each trader has a single opportunity to
trade and equation (31) is the market clearing condition.
The optimal allocation and the maximum gains from trade with a symmetric distribution for the private values with median zero are reported in the
following lemma.
LEMMA 1: Suppose that the private values are drawn from the continuous, symmetric distribution G(· | xt ) with median zero. The allocation that solves (29)
subject to (30) and (31) is
1, for ut ≤ −ce − co
1, for ut ≥ ce + co
sell∗
buy∗
I
(ut ; xt ) =
(ut ; xt ) =
, I
0, else,
0, else.
(32)
The maximum gains from trade are
Maximum gains (xt )
= E I sell∗ (ut ; xt )(−ut − ce − co ) + I buy∗ (ut ; xt )(ut − ce − co ) xt .
(33)
The proof is in Appendix A.
By construction, the current gains from trade in the limit order market
are less than or equal to the maximum gains from trade. The current gains
may be less than the maximum gains because limit orders face execution
risk and the traders’ private incentives may lead them to make order submissions that are different from the ones that would lead to the social optimum.
We decompose the differences between the maximum and current gains into
four sources: no execution, no submission, wrong direction, and extramarginal
submission.
No execution is the expected loss from traders who buy or sell the stock in the
optimal allocation but whose buy or sell orders do not execute in the limit-order
market, that is,
No execution (z t )
⎡
⎤
(ut ; xt )
+ ut ; z t ) 1 −
(−ut − ce ) ⎥
⎢I
⎢
⎥
s=0
⎢
⎥
= E⎢
zt ⎥ .
⎢
⎥
B
⎦
⎣
buy∗
buy
buy∗
+I
(ut ; xt )
d b ( y t + ut ; z t ) 1 − ψb (z t ) (ut − ce )
sell∗
S
d ssell∗ ( y t
ψssell (z t )
(34)
b=0
Losses from no execution arise because it is sometimes individually optimal for
traders with valuations that differ from the common value by more than ce +
co to submit limit orders that may fail to execute.
Estimating the Gains from Trade
2765
No submission is the expected loss from traders who buy or sell the stock in
the optimal allocation but do not submit an order in the limit-order market,
that is,
No submission(z t )
⎤
⎡
S
B
buy∗
⎢ 1−
⎥
d ssell∗ ( y t + ut ; z t ) −
d b ( y t + ut ; z t )
⎢
zt ⎥ .
= E⎣
s=0
b=0
⎦
sell∗
× I
(ut ; xt )(−ut − ce − co ) + I buy∗ (ut ; xt )(ut − ce − co ) (35)
Losses from no submission arise because it is sometimes individually optimal
for traders with valuations that differ from the common value by more than ce
+ co to not submit any order.
Although a trader’s order submission is individually optimal by construction,
it need not lead to a positive contribution to the gains from trade. For example,
suppose a sell market order and a buy limit order transact at price psell
t,0 . If
the seller’s valuation usell
>
0,
the
seller
makes
a
negative
contribution
to
the
t
−
c
−
c
<
0.
Nevertheless,
the
trade
can
be
gains from trade because −usell
e
o
t
individually optimal for the seller if psell
−
y
>
c
+
c
.
In
the
example,
a
trader
t
e
o
t,0
with a positive private value—and hence, no particular need to sell—may end
up selling the stock because the limit-order book provides an opportunity to sell
at a sufficiently high price.
The seller transacts with a buy limit order submitted by a previous trader
with a high valuation; thus the common value at the time of the buy limit
order submission or the previous trader’s private value or both are high.
Depending on the seller’s private value, his trade contributes either to the
wrong direction losses or to the extramarginal submissions that we define
below.
Wrong direction is the expected loss from traders who buy or sell the stock
in the optimal allocation but submit an order to trade in the wrong direction in
the limit-order market:
Wrong direction (z t )
⎤
⎡
B
buy∗
buy
sell∗
(ut ; xt )
d b ( y t + ut ; z t ) −ut − ce + ψb (z t )(−ut + ce ) ⎥
⎢I
⎢
⎥
b=0
⎢
⎥
= E⎢
zt ⎥ .
⎢
⎥
B
⎦
⎣
+ I buy∗ (ut ; xt )
d bsell∗ ( y t + ut ; z t ) ut − ce + ψssell (z t )(ut + ce ) b=0
(36)
Extramarginal submission is the expected loss from traders who do not
trade in the optimal allocation but submit buy or sell orders in the limit-order
market:
The Journal of Finance
2766
Extramarginal submission (z t )
⎤
⎡
1 − I sell∗ (ut ; xt ) − I buy∗ (ut ; xt )
⎢
⎛
⎞ ⎥
S
⎢
⎥
⎢
⎥
d ssell∗ ( y t + ut ; z t ) ψssell (z t )(ut + ce ) + co
⎜
⎟ ⎥
⎢
⎜
⎟
⎢
⎥
= E⎢
⎜ s=0
⎟ zt ⎥ .
⎟ ⎥
⎢ ×⎜
⎜ ⎢
⎥
B
buy
⎟
⎝
⎠ ⎦
⎣
buy∗
+
d b ( y t + ut ; z t ) ψb (z t )(−ut + ce ) + co
(37)
b=0
Another useful comparison to the gains from trade in the limit order market
is the gains from trade that would obtain if bid and ask quotes were made by
a profit-maximizing monopolist with a private value of zero and no inventory
carrying costs. The monopoly gains from trade are determined by finding the
monopolist’s profit-maximizing bid and ask quotes and computing the gains
from trade at the resulting allocation.
m
To describe the monopolist’s problem, let bm
t denote the bid quote and at the
ask quote that the monopolist dealer quotes at time t. To describe a traders’
decision, define the sell indicator function
m
1, if a trader with private value ut sells the stock in state xt
m,sell
bt ; ut ; xt =
I
0, otherwise,
(38)
and define the buy indicator function Im,buy (am
t ; ut ; xt ) similarly.
Using the sell and buy indicators, the monopolist’s profit-maximizing quotes
solve
max
(39)
E I m,sell btm ; ut ; xt y t − btm + I m,bu y atm ; ut ; xt atm − y t ,
m m
{bt ,at }
subject to
E I m,sell btm ; ut ; xt xt = E I m,bu y atm ; ut ; xt xt .
(40)
LEMMA 2: Suppose the private values are drawn from the continuous, symmetric
distribution G(· | xt ) with median zero. The monopolist’s optimal quotes am∗
t and
bm∗
solve
t
G btm∗ − ce − co − y t xt
m∗
bt = y t − m∗
(41)
g bt − ce − co − y t xt
and
atm∗
1 − G atm∗ + ce + co − y t xt
.
= yt +
g atm∗ + ce + co − y t xt
The traders’ decisions given bm∗
and am∗
are
t
t
1, for u ≤ btm∗ − ce − co − y t
I m,sell btm∗ ; ut ; xt =
,
0, otherwise
(42)
(43)
Estimating the Gains from Trade
and
I
m,bu y
m∗
at ; ut ; xt =
1,
for u ≥ atm∗ + ce + co − y t
0,
otherwise
2767
,
(44)
assuming that the second-order conditions reported in equations (A14) and (A15)
are satisfied at the solutions. The monopoly gains from trade are
Monopoly gains (xt )
= E I m,sell btm∗ ; ut ; xt (−ut − ce − co ) + I m,bu y atm∗ ; ut ; xt (ut − ce − co ) xt .
(45)
The proof is in Appendix A.
From equations (41) and (42), bm∗
< yt and am∗
> yt . Comparing the monopot
t
list allocation with the allocation that maximizes the gains from trade in equation (32), the monopolist sets bid and ask quotes that results in less trading
than is efficient.
III. Empirical Results
We use a two-step method to estimate the parameters of the model. In the first
step, we use the execution and cancellation histories of the order submissions
to estimate the execution probabilities and picking-off risks. In the second step,
we estimate the private value distributions, traders’ arrival rates, and costs by
maximizing the conditional log-likelihood function for limit and market order
arrival times. We use the estimated parameters to form estimates of the current,
maximum, and monopoly gains from trade.
A. Description of the VSE and Our Sample
Our sample comes from the audit tapes of the VSE. The limit order trading
system used by the VSE is very similar to the systems used by the London
Stock Exchange, Euronext, the Stockholm Stock Exchange, the Toronto Stock
Exchange, the Archipelago Exchange and many other markets. In 1999, after
the end of our sample, the VSE was involved in an amalgamation of Canadian
equity trading and became part of the Canadian Venture Exchange, which in
turn was recently renamed the TSX Venture Exchange. The TSX Venture Exchange is similar in structure, rules, and activity to the VSE.
Forty-five exchange member firms act as brokers, submitting orders for outside traders, and as dealers, submitting orders on their own account. There are
no designated market makers. The market is open from 6:30 a.m. to 1:30 p.m.
Pacific time. Limit orders in the order book are matched with incoming market
orders to produce trades, giving priority to limit orders according to the order
price and then the time of submission. Order prices must be multiples of a tick
size. The tick size varies between one cent for prices below $3.00, five cents for
prices between $3.00 and $4.99, and twelve-and-a-half cents for prices at $5.00
2768
The Journal of Finance
Table I
Order Submissions
The sample period is May 1, 1990, to November 30, 1993.
Barkhor Resources
BHO
Eurus Resources
ERR
War Eagle
Mining Company
WEM
55,444
56,599
47,578
31.7
21.6
28.5
18.2
31.5
23.7
27.9
16.9
31.3
19.9
32.0
16.8
Time between order submissions in seconds
Average
247.0
Standard deviation
842.4
297.9
983.7
396.6
1089.4
Stock
Ticker
Number of order submissions
Percent of submissions
Sell limit orders
Sell market orders
Buy limit orders
Buy market orders
and above. Orders sizes must be multiples of a fixed size which varies between
100 and 1,000 shares.
Member firms can also submit hidden limit orders, where a fraction of the
order size is not visible on the limit order book. The hidden fraction of the order
retains its price priority, but loses its time priority. In our sample, few hidden
orders are submitted, thus the assumption of no hidden limit orders in our
model is a reasonable approximation for our sample.
Our sample contains a record for every trade, cancellation, or change in the
status of an order, and the limit-order book at the open of each day. Each record
includes the time of the original order submission, but not the member firms’
identification codes nor whether or not a member firm submitted an order as a
broker or dealer. Combining the records with the limit-order book at the open
of each day we reconstruct order histories for each order submission, including
the initial order submission and every future order execution or cancellation,
and the corresponding order books. Less than 1% of the orders are associated
with inconsistencies between the inferred order histories and the trading rules.
We drop such orders from our sample.
Our sample goes from May 1990 to November 1993 for three stocks in the
mining industry. Table I reports the name and ticker symbol of the three stocks.
The table reports the total number of order submissions, the percentage of buy
and sell market and limit orders submitted in our sample, and the average and
standard deviation of the time between order submissions.
B. Construction of the Variables
We use a centered moving average of the mid-quotes over a 20-minute window
to proxy for the stock’s unobserved common value. Our proxy is reasonable
because most of the time the best quotes should straddle the common value.
Estimating the Gains from Trade
2769
We use a centered moving average to reduce the impact of mechanical shifts in
the mid-quote caused by individual order submissions or cancellations.
Table II reports our choice for the state vector zt = (xt , wt ). The table reports
the names of the variables, a brief description of them, and their sample means
and standard deviations. In the theoretical model, the exogenous state variables, xt , predict the trader arrival rates, the distribution of innovations to the
common value, and the conditional distributions of the traders’ private values. Our choice of exogenous state variables is reported in the top panel of
Table II. We observe the exogenous state variables at a daily frequency.
We select exogenous state variables that are likely to be correlated with the
traders’ desire to change their portfolios and with innovations in the stocks’
common value. In particular, the exogenous state variables that we use are
Toronto Stock Exchange (TSX) market index volatility, TSX mining volatility,
interest rate volatility, exchange rate volatility, and stock volatility. Traders
may be more likely to want to change their portfolio as a result of changes in
the market index, interest rates, or the stock price. For example, a change in
the market index may lead more traders to wish to change their portfolios and
may also change the traders’ willingness to pay more to buy immediately or
receive less to sell immediately. Such effects are captured by changes in the
trader arrival rate and the distribution of private values.
The bottom panel in Table II reports our choice of endogenous state variables,
wt . Ideally, wt would include the entire limit-order book and any other variables
known at t that are useful for predicting the outcomes of order submissions at t.
However, because our sample size is relatively small we use a smaller number
of variables in wt .
The bid–ask spread and measures of depth close to the quotes and away from
the quotes directly measure the state of the limit order book. Close depth is the
number of shares outstanding at the current best quotes and far depth is the
cumulative number of shares outstanding up to and including the marginal
limit order. The depth measures on the same side of the book often contain
very similar information. We therefore in our first-step estimation include only
one depth measure for each side of the market. We use the depth in front
of the order as well as the close depth on the other side of the book to predict the execution probabilities and picking-off risks. Execution probabilities
depend on book variables through the length of the order queues at different prices. Execution probabilities also depend on book variables indirectly
through the book’s effect on the current and future order submissions and
cancellations.
Past order submission activity is useful to predict the latent execution and
cancellation times for new order submissions because the average age of existing limit orders in the book can inf luence the probability that the existing
limit orders are canceled. We use the number of recent trades and lagged durations to measure past order submission activity. Holding everything else equal,
larger orders are likely to have lower execution probabilities and to face higher
picking-off risk. The distance between the current mid-quote and our proxy
for the common value is included because holding everything else equal, a buy
2770
The Journal of Finance
Table II
The State Vector, zt
The table describes the construction of the state vector zt = (xt , wt ), with xt , the exogenous state
variables, and wt , the endogenous state variables. We also report the means and standard deviations
of all the variables for the three stocks in our sample: Barkhor Resources (BHO), Eurus Resources
(ERR), and War Eagle Mining Company (WEM).
BHO
Name
TSX market
volatility
TSX mining
volatility
Interest rate
volatility
Exchange rate
volatility
Stock
volatility
Spread
Close ask
depth
Far ask depth
Close bid
depth
Far bid depth
Order
quantity
Recent trades
Lagged
duration
Mid-quote
volatility
Distance to
mid-quote
First hour
Second hour
Third hour
Fourth hour
Fifth hour
Sixth hour
Description
Mean
SD
Exogenous State Variables: xt
Absolute value of the lagged
0.39
0.35
close-to-close return on the TSX
market index
Absolute value of the lagged
1.29
1.19
close-to-close return on the TSX
mining index
Absolute value of the lagged change 0.16
0.21
in the overnight interest rate
Absolute value of the lagged change 0.20
0.19
in the Canadian/U.S. exchange
rate
Absolute value of the lagged
4.28
8.89
open-to-open stock return
Endogenous State Variables: wt
Bid–ask spread
0.02
0.02
Logarithm of ask depth at best ask
1.74
0.95
Logarithm of cumulative ask depth
up to the marginal sell order
Logarithm of bid depth at best bid
ERR
WEM
Mean
SD
Mean
SD
0.39
0.35
0.39
0.34
1.30
1.19
1.29
1.19
0.16
0.21
0.16
0.21
0.20
0.18
0.20
0.19
4.16
6.11
3.59
5.04
0.05
0.94
0.06
0.79
0.04
1.02
0.03
0.75
4.04
0.83
2.60
0.81
2.83
0.69
1.75
1.00
1.04
0.85
1.08
0.83
Logarithm of cumulative bid depth
4.06
down to the marginal buy order
Logarithm of the number of shares
1.49
in the order
Number of trades in the last 10
8.61
minutes
Sum of last 10 durations of order
3.10
book changes, divided by 1,000
Volatility of the mid-quote over the
1.71
last 10 minutes
Moving average mid-quote minus
0.00
the mid-quote, measured as a
percentage
Time Dummies
Dummy variable for 6:30–7:30
Dummy variable for 7:30–8:30
Dummy variable for 8:30–9:30
Dummy variable for 9:30–10:30
Dummy variable for 10:30–11:30
Dummy variable for 11:30–12:30
0.78
2.82
0.85
3.14
0.77
0.83
0.72
0.90
0.84
0.83
10.88
5.06
6.75
4.02
5.68
5.30
3.16
4.96
3.93
5.20
0.07
1.71
0.09
1.70
0.05
0.02
0.00
0.04
0.00
0.02
Estimating the Gains from Trade
2771
order at two ticks below the common value is less likely to be executed than a
buy order one tick below the common value.
We include six hourly dummy variables to capture any deterministic timeof-day patterns in execution probabilities and picking-off risks. Deterministic
time-of-day patterns may arise because of deadline effects associated with market closure. For example, some traders cancel unexecuted limit orders at the
close of the market. Such behavior introduces time-of-day patterns in the timing
of cancellations, as orders submitted early are less likely to remain outstanding
at the time of the market close than orders submitted later.
We assume that some traders always find it optimal to submit one-tick buy
and sell limit orders. In the theoretical model, the marginal sell limit order is
defined as the highest-priced sell limit order that any trader would optimally
submit at zt . The marginal buy limit order is defined similarly. Empirically, we
set the marginal prices depending on the level of the common value. For each
decile of the common value, we define a cutoff sell price as the price such that at
least 95% of the sell order prices are below that cutoff sell price. The marginal
sell order is defined as the lowest-priced sell order above the cutoff price. The
marginal buy order is defined similarly.
We purposely drop 5% of the order submissions to avoid a situation in which
the orders submitted at extreme prices—which may represent order entry
mistakes—would determine the marginal order prices. By ignoring limit orders
submitted outside the price range defined by the marginal prices, we ignore the
expected payoffs received by traders whose private values would lead them to
submit limit orders outside that price range. Omitting some payoffs may lead
to a downward bias in our estimates of the current gains from trade, but should
not affect our estimates of the maximum gains from trade and the monopoly
gains from trade.
We need to estimate the execution hazard rate, the cancellation hazard rate,
and the expected change in the common value conditional on execution for the
marginal limit orders. By construction, the marginal limit orders are a small
fraction of the orders submitted and tend to have a low execution probability.
In estimating the execution hazard rate, the cancellation hazard rate, and the
expected change in the common value conditional on execution for the marginal
limit orders, we combine orders submitted at the marginal price with orders
submitted up to two ticks away from the marginal price. By grouping orders, we
ensure that we have enough executed orders to obtain estimates of the hazard
rates and the expected change in the common value conditional on execution
for the marginal limit orders.
In the theoretical model, order quantity is normalized to one unit: All orders
are either fully executed or canceled. In our sample, however, different order
submissions have different order quantities, that is, partial executions occur.
Empirically, we handle partial executions by assuming an order was fully executed if at least 50% of the order size is executed and fully cancelled otherwise.
Table III reports the average percentage of the submitted limit order quantity
that is executed within 48 hours, conditional on that percentage being at least
50% or less than 50%. Less than 1% of the order executions occur more than
The Journal of Finance
2772
Table III
Order Executions and Cancellations
The top panel reports for all limit orders the average percentage of the submitted order quantity
that is executed within 48 hours of order submission, conditional on that percentage being at least
50% or less than 50%. Standard errors are reported in parentheses. The second panel reports, for all
limit orders that eventually execute, the percent that execute within five different time intervals.
For orders that have several partial executions, the time of execution is weighted by the quantity
executed to determine an average execution time. The third panel reports, for all orders that are
eventually cancelled, the percent that cancel within five time intervals relative to the time of the
order submission, the percent that cancel within five time intervals before the market close on the
day of order submission, and the percent that cancel the next day or later. The three sample stocks
are Barkhor Resources (BHO), Eurus Resources (ERR), and War Eagle Mining Company (WEM).
BHO
ERR
WEM
96.97
(0.07)
1.43
(0.04)
96.58
(0.09)
1.21
(0.04)
Distribution of Time to Execution
Percent of orders executed
Less than 5 minutes after order submission
46.67
5 to 15 minutes after order submission
15.92
15 minutes to 1 hour after order submission
19.59
1 to 3 hours after order submission
10.73
More than 3 hours after order submission
7.09
42.77
17.97
21.92
11.46
5.88
37.29
17.24
22.60
14.07
8.80
Distribution of Time to Cancellation
Percent of orders cancelled
Less than 5 minutes after order submission
31.79
5 to 15 minutes after order submission
13.08
15 minutes to 1 hour after order submission
16.42
1 to 3 hours after order submission
14.30
More than 3 hours after order submission
24.41
27.57
13.81
18.17
15.45
25.00
23.49
11.78
16.64
17.70
30.39
Percent of orders cancelled
Less than 5 minutes before market close
5 to 15 minutes before market close
15 minutes to 1 hour before market close
1 to 3 hours before market close
More than 3 hours before market close
After the first day market close
21.16
2.51
8.68
20.31
41.12
6.22
26.32
2.80
9.46
20.35
33.05
8.02
At least 50% executes
Less than 50% executes
Conditional Order Execution Percentage
97.34
(0.07)
1.40
(0.04)
18.77
2.84
9.15
20.80
40.78
7.66
48 hours from the time of the order submission. The average execution percentage is close to 100% conditional on the execution percentage being 50% or more,
and the average execution percentage is close to 0% conditional on the execution percentage being less than 50%. Our assumption with respect to partial
execution is therefore a reasonable approximation of our sample.
In the theoretical model, both execution and cancellation times are random.
The assumption of random execution and cancellation times is also justified
in our sample. The second panel of Table III, which reports the distribution
Estimating the Gains from Trade
2773
of the time to execution for all limit orders in our sample shows that time to
execution is indeed random. Similarly, the third panel of Table III shows that
the time to cancellation is also random; orders are not canceled a fixed number
of minutes after submission nor are they all canceled at the end of the trading
day.
C. Estimates of the Execution Probabilities and Picking-Off Risks
We assume that the traders’ beliefs about the execution probabilities and
picking-off risks are consistent with the empirical execution and cancellation
histories.
The execution probabilities are determined by the distributions of the latent
times to cancellation and execution in equations (5) and (6). The picking-off
risks are determined by the execution probabilities and the expected change
in the common value conditional on the order executing. We use our sample to
estimate both the distribution of the latent times and the expected change in
the common value conditional on the order executing. We then use the resulting
estimates to compute estimates of execution probabilities and picking-off risks.
In turn, the estimates of the execution probabilities and picking-off risks are
used to characterize the traders’ expectations and the optimal order submission
strategy.
Our formulation of independent latent execution and cancellation times is a
competing risks model. Appendix B provides a brief description of the competing
risks model. We use the distribution of the latent execution and cancellation
times to compute the execution probabilities. Our approach extends Cho and
Nelling (2000), who compute execution probabilities for limit orders based on
the parameter estimates for the distribution of the time to execution, assuming
that all orders are canceled at the end of the day.
We parameterize the conditional distributions of the cancellation times as a
Weibull,
buy
buy buy Fcancel τ z t , d t,b = 1 = 1 − exp −exp z t γb τ αb ,
(46)
where zt is the state vector. The hazard rate is defined as the probability that
the cancellation time occurs between t + τ and t + τ + dτ , conditional on the
cancellation time being greater than t + τ . For the Weibull model, the hazard
rate is
buy
buy buy buy
Pr τcancel ∈ [τ, τ + d τ ) τcancel ≥ τ, z t , d t,b = 1 = exp z t γb αb τ αb −1 d τ.
(47)
buy
The parameter vector γb measures the effect of the state vector on the hazard
rate. If a variable has a positive parameter, then an increase in that variable
buy
increases the hazard rate. The parameter αb is the Weibull shape parameter.
buy
buy
If αb = 1, the hazard rate does not depend on τ . If αb < 1, the hazard rate
buy
is decreasing in τ . If αb > 1, the hazard rate is increasing in τ .
2774
The Journal of Finance
Table IV reports the results for the cancellation time distributions. The models are estimated for one-tick and marginal limit orders and are estimated by
maximum likelihood. We treat orders that last longer than 2 days as censored
observations.
The parameter estimates for the Weibull shape parameters are all less than
1, with an average value of 0.56. The cancellation hazard rates are decreasing in
the time that the order is in the book. The age of an order predicts the conditional
probability of cancellation. Past activity is correlated with the age of unfilled
orders in the book. Past activity therefore can predict the cancellation rates of
existing orders in the book, consistent with the assumption in the theoretical
model.
We also parameterize the conditional distributions of the time to execution
as a Weibull
buy buy buy
Fexecute τ z t , d t,b = 1 = 1 − exp −exp z t κb τ βb .
(48)
The conditional distributions of the time to execution depend on the trader
arrival rates, the order cancellation distributions, and future traders’ order
submissions. A disadvantage of the parametric model is that it imposes auxiliary restrictions on the conditional distributions. An alternative, which does
not impose such auxiliary assumptions, is to use nonparametric methods as in
Hollifield et al. (2004). An advantage of the parametric model is that we can use
a larger state vector than with a nonparametric method. We therefore choose
the parametric model because it allows us to approximate the large information
set available to the traders.
Table V reports the results for the execution time distributions. The models
are estimated for one-tick and marginal limit orders and are estimated by maximum likelihood. We treat the orders that last longer than 2 days as censored
observations.
The parameter estimates for the Weibull shape parameters are all less than
1, with an average value of 0.65. The execution hazard rates are decreasing in
the time that the order is in the book. The Weibull shape parameter is lower
for the cancellation times than for the execution times; the probability that a
limit order is canceled rather than executed decreases with the time the order
is in the book.
Tables IV and V also report chi-squared tests for the null hypotheses that the
conditional distributions of the execution and cancellation times do not depend
on the state vector zt . We reject the null hypothesis for all order submissions
and stocks.
We use the parameter estimates from the conditional distributions of the
execution times and cancellation times to forecast the execution probabilities
for buy and sell one-tick and marginal limit orders at every order submission.
We compute the probability that the order executes within 2 days: T = 2 days.
Appendix C reports details of the computations of the execution probabilities.
For Barkhor Resources (BHO), the average execution probability for marginal
sell limit orders is approximately 16%, for one-tick sell limit orders is 61%, for
Table IV
0.63
(0.02)
−5.08
(0.89)
0.01
(0.03)
0.06
(0.03)
0.00
(0.03)
−0.02
(0.03)
0.00
(0.03)
−0.54
(1.48)
—
—
−0.15
(0.04)
−0.01
(0.03)
Shape
Close ask
depth
Far ask
depth
Close bid
depth
TSX market
volatility
TSX mining
volatility
Interest rate
volatility
Exchange rate
volatility
Stock
volatility
Spread
Constant
Marg.
Variable
0.51
(0.02)
−1.72
(0.85)
0.02
(0.03)
0.03
(0.02)
0.01
(0.02)
0.02
(0.02)
0.07
(0.02)
−9.19
(4.21)
−0.01
(0.03)
—
—
0.00
(0.02)
1 Tick
Sell Orders
Marg.
0.52
(0.01)
−6.93
(0.73)
−0.00
(0.03)
0.00
(0.03)
−0.03
(0.02)
−0.05
(0.03)
0.09
(0.03)
2.20
(4.33)
0.05
(0.03)
—
—
−0.01
(0.03)
1 Tick
Buy Orders
0.62
(0.02)
−6.03
(0.63)
−0.01
(0.03)
0.01
(0.03)
0.02
(0.03)
−0.06
(0.03)
0.07
(0.03)
3.46
(1.69)
0.01
(0.04)
—
—
—
—
BHO
0.62
(0.01)
−6.64
(0.98)
0.05
(0.02)
−0.00
(0.03)
0.05
(0.02)
−0.03
(0.03)
−0.00
(0.02)
2.21
(0.71)
—
—
−0.06
(0.03)
−0.01
(0.03)
Marg.
0.50
(0.01)
−5.60
(1.21)
0.00
(0.03)
0.01
(0.03)
0.05
(0.03)
−0.04
(0.03)
−0.06
(0.04)
−5.59
(2.22)
0.00
(0.04)
—
—
0.02
(0.03)
1 Tick
Sell Orders
Marg.
0.54
(0.02)
−2.76
(0.81)
0.05
(0.04)
−0.12
(0.04)
0.05
(0.05)
−0.08
(0.04)
0.01
(0.03)
0.21
(2.70)
−0.05
(0.04)
—
—
−0.02
(0.04)
1 Tick
Buy Orders
0.59
(0.01)
−4.23
(0.66)
0.04
(0.03)
−0.05
(0.03)
−0.00
(0.03)
−0.02
(0.03)
−0.03
(0.03)
2.71
(0.63)
−0.05
(0.03)
—
—
—
—
ERR
0.62
(0.01)
−4.51
(0.91)
−0.01
(0.03)
0.01
(0.03)
0.03
(0.03)
−0.07
(0.03)
0.01
(0.03)
3.54
(1.08)
—
—
−0.15
(0.04)
−0.05
(0.04)
Marg.
0.49
(0.01)
1.11
(1.76)
0.01
(0.04)
0.07
(0.04)
−0.00
(0.04)
−0.01
(0.04)
0.00
(0.04)
3.64
(3.02)
−0.08
(0.05)
—
—
0.08
(0.04)
1 Tick
Sell Orders
Marg.
(continued)
0.47
(0.01)
−9.61
(1.96)
0.03
(0.04)
−0.01
(0.04)
0.00
(0.04)
−0.03
(0.05)
0.07
(0.04)
−7.16
(3.95)
0.04
(0.05)
—
—
−0.08
(0.04)
1 Tick
Buy Orders
0.65
(0.01)
−3.55
(1.09)
0.01
(0.03)
0.00
(0.03)
−0.03
(0.03)
−0.06
(0.03)
0.04
(0.03)
2.11
(0.83)
−0.02
(0.04)
—
—
—
—
WEM
Parameter estimates with asymptotic standard errors in parentheses for a Weibull model for the cancellation times. The χ 2 test is for the null that
the state vector zt does not affect the conditional distribution with p-values in parentheses. The three sample stocks are Barkhor Resources (BHO),
Eurus Resources (ERR), and War Eagle Mining Company (WEM).
Weibull Model for Cancellation Times
Estimating the Gains from Trade
2775
No. of Obs.
χ 2 (19)
p-value
Sixth hour
Fifth hour
Fourth hour
Third hour
Second hour
Far bid
depth
Order
quantity
Recent
trades
Lagged
duration
Mid-quote
volatility
Distance to
mid-quote
First hour
Variable
1,748
—
—
0.04
(0.04)
0.01
(0.00)
−0.11
(0.01)
0.10
(0.49)
0.22
(2.08)
−0.67
(0.11)
−0.66
(0.12)
−0.37
(0.12)
−0.35
(0.12)
−0.05
(0.12)
0.13
(0.12)
290.01
(0.00)
Marg.
4,498
—
—
−0.05
(0.03)
0.03
(0.00)
−0.02
(0.00)
−1.09
(0.50)
21.25
(2.19)
−1.12
(0.09)
−0.96
(0.08)
−0.90
(0.08)
−0.86
(0.09)
−0.64
(0.08)
−0.46
(0.08)
576.14
(0.00)
1 Tick
Sell Orders
4,105
—
—
−0.12
(0.03)
0.03
(0.00)
−0.05
(0.01)
1.99
(0.43)
−11.92
(2.12)
−1.01
(0.09)
−1.11
(0.09)
−1.00
(0.09)
−0.77
(0.08)
−0.89
(0.08)
−0.65
(0.09)
648.10
(0.00)
−0.08
(0.04)
0.03
(0.04)
0.02
(0.00)
−0.08
(0.01)
0.71
(0.36)
−2.83
(1.73)
−1.04
(0.13)
−0.87
(0.13)
−0.76
(0.14)
−0.77
(0.14)
−0.43
(0.14)
−0.25
(0.15)
247.37
(0.00)
1,353
1 Tick
Buy Orders
Marg.
BHO
2,458
—
—
0.14
(0.03)
0.02
(0.00)
−0.06
(0.01)
1.04
(0.56)
9.41
(2.93)
−1.07
(0.10)
−0.97
(0.10)
−0.91
(0.10)
−0.61
(0.10)
−0.77
(0.10)
−0.38
(0.10)
258.47
(0.00)
Marg.
2,368
—
—
−0.03
(0.03)
0.01
(0.01)
−0.01
(0.01)
1.53
(0.71)
25.91
(5.05)
−0.82
(0.11)
−1.03
(0.11)
−0.97
(0.12)
−0.82
(0.11)
−0.63
(0.11)
−0.49
(0.11)
211.25
(0.00)
1 Tick
Sell Orders
2,138
−0.07
(0.03)
0.01
(0.03)
0.03
(0.01)
−0.06
(0.01)
−0.08
(0.38)
−9.10
(2.47)
−1.09
(0.10)
−1.02
(0.10)
−0.94
(0.10)
−0.77
(0.11)
−0.64
(0.11)
−0.37
(0.11)
304.15
(0.00)
1,732
—
—
−0.02
(0.04)
0.04
(0.01)
−0.04
(0.01)
−0.24
(0.46)
−31.00
(6.37)
−0.72
(0.13)
−0.81
(0.13)
−1.09
(0.14)
−1.02
(0.13)
−0.77
(0.13)
−0.77
(0.13)
218.13
(0.00)
1 Tick
Buy Orders
Marg.
ERR
Table IV—Continued
1,679
—
—
0.06
(0.04)
0.04
(0.01)
−0.05
(0.01)
−0.17
(0.52)
7.16
(2.54)
−1.02
(0.11)
−1.01
(0.11)
−0.73
(0.11)
−0.74
(0.11)
−0.50
(0.12)
−0.33
(0.11)
273.52
(0.00)
Marg.
1,793
—
—
−0.06
(0.05)
0.04
(0.01)
−0.03
(0.01)
−2.74
(1.03)
31.78
(3.91)
−0.96
(0.14)
−0.80
(0.13)
−0.87
(0.13)
−0.73
(0.13)
−0.56
(0.13)
−0.41
(0.12)
188.44
(0.00)
1 Tick
Sell Orders
1,792
−0.09
(0.04)
−0.05
(0.03)
0.03
(0.01)
−0.05
(0.01)
−0.95
(0.64)
−5.01
(2.73)
−1.03
(0.11)
−0.69
(0.10)
−0.71
(0.11)
−0.48
(0.11)
−0.33
(0.12)
−0.12
(0.11)
273.48
(0.00)
1,689
—
—
−0.10
(0.04)
0.04
(0.01)
−0.03
(0.01)
3.84
(1.15)
−12.41
(5.57)
−0.62
(0.16)
−1.07
(0.15)
−0.68
(0.12)
−0.55
(0.12)
−0.85
(0.13)
−0.49
(0.12)
201.66
(0.00)
1 Tick
Buy Orders
Marg.
WEM
2776
The Journal of Finance
Table V
Close ask
depth
Far ask
depth
Close bid
depth
TSX market
volatility
TSX mining
volatility
Interest rate
volatility
Exchange rate
volatility
Stock
volatility
Spread
Constant
Shape
Variable
0.66
(0.03)
−15.69
(1.51)
0.07
(0.05)
−0.03
(0.07)
−0.03
(0.07)
0.10
(0.06)
0.15
(0.06)
−1.53
(2.69)
—
—
−0.28
(0.08)
−0.23
(0.07)
Marg.
0.58
(0.01)
−10.42
(0.42)
−0.00
(0.02)
−0.06
(0.02)
−0.02
(0.02)
0.02
(0.02)
0.10
(0.02)
56.84
(2.36)
−0.06
(0.02)
—
—
0.14
(0.02)
1 Tick
Sell Orders
BHO
0.70
(0.04)
−4.31
(1.85)
0.05
(0.08)
0.01
(0.08)
0.06
(0.09)
−0.38
(0.10)
0.09
(0.08)
16.29
(2.78)
0.00
(0.10)
—
—
—
—
Marg.
0.61
(0.01)
2.04
(0.47)
0.02
(0.02)
−0.04
(0.02)
0.02
(0.02)
−0.04
(0.02)
0.06
(0.02)
68.73
(2.67)
0.20
(0.02)
—
—
−0.08
(0.02)
1 Tick
Buy Orders
0.65
(0.02)
−12.97
(0.89)
0.18
(0.04)
−0.16
(0.05)
−0.09
(0.05)
−0.10
(0.05)
0.01
(0.04)
3.46
(1.29)
—
—
−0.21
(0.06)
0.09
(0.05)
Marg.
0.54
(0.01)
−10.35
(0.58)
0.03
(0.03)
0.01
(0.03)
−0.08
(0.04)
−0.07
(0.04)
0.00
(0.04)
22.51
(1.57)
−0.28
(0.04)
—
—
0.20
(0.03)
1 Tick
Sell Orders
ERR
0.82
(0.03)
−3.86
(1.23)
0.00
(0.05)
0.01
(0.05)
0.08
(0.04)
−0.07
(0.05)
0.07
(0.03)
3.65
(1.20)
0.11
(0.06)
—
—
—
—
Marg.
0.64
(0.02)
3.93
(1.86)
−0.03
(0.03)
−0.02
(0.03)
−0.02
(0.05)
−0.01
(0.04)
0.03
(0.03)
25.82
(1.81)
0.07
(0.04)
—
—
−0.08
(0.04)
1 Tick
Buy Orders
0.64
(0.03)
−5.93
(1.60)
−0.04
(0.06)
−0.26
(0.07)
0.02
(0.05)
−0.01
(0.06)
−0.05
(0.06)
3.98
(2.13)
—
—
−0.43
(0.08)
0.02
(0.07)
Marg.
0.58
(0.01)
−26.41
(1.94)
0.04
(0.04)
−0.04
(0.04)
0.01
(0.04)
−0.05
(0.05)
0.10
(0.03)
19.35
(1.43)
−0.02
(0.04)
—
—
0.17
(0.04)
1 Tick
Sell Orders
0.59
(0.01)
12.91
(2.01)
0.00
(0.04)
0.05
(0.04)
0.09
(0.03)
0.03
(0.04)
0.10
(0.04)
35.60
(2.53)
0.05
(0.05)
—
—
−0.20
(0.04)
(continued)
0.79
(0.04)
1.22
(2.01)
0.07
(0.06)
0.10
(0.06)
0.06
(0.06)
−0.14
(0.07)
0.15
(0.06)
3.22
(1.68)
0.02
(0.08)
—
—
—
—
1 Tick
Buy Orders
Marg.
WEM
Parameter estimates, with asymptotic standard errors in parentheses, for a Weibull model for the execution times. The χ 2 test is for the null that
the state vector zt does not affect the conditional distribution, with p-values in parentheses. The three sample stocks are Barkhor Resources (BHO),
Eurus Resources (ERR), and War Eagle Mining Company (WEM).
Weibull Model for Execution Times
Estimating the Gains from Trade
2777
No. of Obs.
χ 2 (19)
p-value
Sixth hour
Fifth hour
Fourth hour
Third hour
1,748
—
—
−0.19
(0.08)
0.03
(0.01)
−0.14
(0.03)
5.61
(0.80)
26.97
(2.76)
−0.68
(0.27)
−0.20
(0.27)
−0.02
(0.27)
0.06
(0.28)
0.28
(0.28)
0.40
(0.29)
302.61
(0.00)
Far bid
depth
Order
quantity
Recent
trades
Lagged
duration
Mid-quote
volatility
Distance to
mid-quote
First hour
Second hour
Marg.
Variable
4,498
—
—
−0.22
(0.03)
0.04
(0.00)
−0.08
(0.01)
3.33
(0.24)
38.51
(1.83)
−0.57
(0.09)
−0.63
(0.09)
−0.44
(0.09)
−0.41
(0.09)
−0.34
(0.09)
−0.21
(0.09)
1731.08
(0.00)
1 Tick
Sell Orders
1,353
0.05
(0.10)
−0.50
(0.10)
0.03
(0.01)
−0.07
(0.03)
−2.17
(1.08)
−21.66
(3.45)
0.08
(0.39)
−0.23
(0.39)
−0.51
(0.43)
−0.28
(0.44)
−0.46
(0.48)
0.00
(0.49)
188.96
(0.00)
4,105
—
—
−0.29
(0.03)
0.04
(0.00)
−0.06
(0.01)
−4.31
(0.27)
−45.17
(1.98)
−0.21
(0.09)
−0.11
(0.09)
−0.26
(0.09)
−0.19
(0.09)
−0.17
(0.09)
0.01
(0.09)
1710.49
(0.00)
1 Tick
Buy Orders
Marg.
BHO
2,458
—
—
−0.12
(0.05)
0.01
(0.01)
−0.15
(0.02)
4.05
(0.47)
36.36
(4.65)
−0.55
(0.20)
−0.55
(0.20)
−0.42
(0.20)
−0.10
(0.20)
−0.11
(0.20)
−0.19
(0.22)
290.14
(0.00)
Marg.
2,368
—
—
−0.02
(0.03)
0.02
(0.01)
−0.09
(0.02)
3.68
(0.32)
49.49
(4.64)
−0.35
(0.12)
−0.48
(0.12)
−0.34
(0.13)
−0.11
(0.12)
−0.11
(0.13)
−0.23
(0.13)
531.01
(0.00)
1 Tick
Sell Orders
1,732
—
—
−0.19
(0.04)
0.01
(0.01)
−0.08
(0.01)
−4.96
(1.08)
−101.78
(6.04)
0.06
(0.13)
−0.17
(0.13)
−0.13
(0.13)
−0.33
(0.13)
−0.22
(0.14)
−0.05
(0.13)
539.69
(0.00)
−0.26
(0.06)
−0.25
(0.06)
0.06
(0.01)
−0.10
(0.02)
−2.53
(0.70)
−45.87
(4.46)
−0.29
(0.24)
−0.20
(0.24)
−0.29
(0.25)
−0.12
(0.25)
−0.10
(0.26)
−0.17
(0.28)
403.44
(0.00)
2,138
1 Tick
Buy Orders
Marg.
ERR
Table V—Continued
1,679
—
—
−0.20
(0.08)
0.04
(0.01)
−0.08
(0.02)
0.28
(0.91)
25.43
(2.62)
−0.92
(0.22)
−1.03
(0.22)
−0.48
(0.22)
−0.06
(0.21)
−0.14
(0.24)
−0.08
(0.22)
218.20
(0.00)
Marg.
1,793
—
—
−0.13
(0.05)
0.04
(0.01)
−0.06
(0.01)
12.89
(1.14)
42.53
(3.72)
−0.81
(0.15)
−0.78
(0.14)
−0.48
(0.13)
−0.44
(0.13)
−0.46
(0.14)
−0.13
(0.13)
501.50
(0.00)
1 Tick
Sell Orders
1,792
−0.49
(0.07)
−0.30
(0.07)
0.06
(0.01)
−0.07
(0.02)
−5.37
(1.18)
−31.16
(5.15)
0.05
(0.31)
0.02
(0.32)
0.39
(0.32)
−0.20
(0.34)
0.63
(0.33)
0.45
(0.33)
244.62
(0.00)
1,689
—
—
−0.20
(0.04)
0.04
(0.01)
−0.04
(0.01)
−10.44
(1.18)
−81.30
(5.88)
−0.20
(0.16)
−0.10
(0.14)
−0.27
(0.13)
−0.21
(0.13)
−0.19
(0.14)
−0.07
(0.13)
493.20
(0.00)
1 Tick
Buy Orders
Marg.
WEM
2778
The Journal of Finance
Estimating the Gains from Trade
2779
marginal buy limit orders is 13%, and for one-tick buy limit orders is 63%. The
estimates for the other stocks are similar.
From equation (11), the picking-off risk is equal to the product of the expected
change in the common value conditional on an execution and the execution
probability. For buy orders, we parameterize the expected change conditional
on an execution as a linear function, that is,
buy
buy
E ( y t+τexecute − y t ) | It (τexecute ≤ τcancel ) = 1, z t , d t,b = 1 = z t b ,
(49)
buy
where zt is the state vector and b is a coefficient vector to be estimated.
The sell side is treated similarly. The expectation of the change in the common
value conditional on execution is determined by the trader arrival rates, the
cancellation time distributions, and the future traders’ order submissions. As
in the case of the distribution of execution times, we use a parametric model
rather than a nonparametric model to allow for a large state vector.
We estimate the model in equation (49) for buy and sell one-tick and marginal
limit orders that execute in our sample using ordinary least squares. Table VI
reports the estimates. The table also reports F-tests for the null hypothesis that
the expected change in the common value conditional on the order executing
does not depend on the state vector zt . We reject the null hypothesis for all order
submissions and stocks.
We use the parameter estimates to forecast the expected change in the common value, conditional on the limit order executing for buy and sell one-tick
and marginal limit orders at every order submission. At the mean values of the
state vector, the expected change in the common value is approximately zero for
one-tick limit orders, minus 4 cents for marginal buy limit orders, and 4 cents
for marginal sell limit orders.
We form estimates of the picking-off risk by substituting our estimates of
the expected change in the common value conditional on execution and the
execution probabilities into equation (11). At the mean values of the state vector,
the picking-off risk is close to zero for one-tick limit orders, and approximately
1 cent for marginal limit orders.
D. Estimates of the Arrival Rates, Private Value Distributions, and Costs
We estimate the remaining parameters of the model by maximizing the conditional log-likelihood function for the timing of market and limit orders. We
form the log-likelihood function for sell market orders, sell limit orders between
the one-tick and the marginal sell order, buy limit orders between the one-tick
and the marginal buy order, and buy market orders. The grouping is consistent
with the theoretical model and leads to consistent estimators of the remaining
parameters. To form the conditional log-likelihood function, we use the optimal order submission strategy, the trader arrival rates, and the distributions
of the trader’s private values to compute the probabilities of observing a limit
order or a market order. We report the conditional log-likelihood function in
Appendix D.
Table VI
7.30
(2.73)
−0.05
(0.10)
0.36
(0.12)
−0.25
(0.12)
−0.02
(0.12)
−0.10
(0.15)
3.66
(6.27)
—
—
0.28
(0.16)
−0.10
(0.12)
—
—
Constant
Close ask
depth
Far ask
depth
Close bid
depth
Far bid
depth
TSX market
volatility
TSX mining
volatility
Interest rate
volatility
Exchange rate
volatility
Stock
volatility
Spread
Marg.
Variable
Marg.
−6.66
(4.87)
0.10
(0.15)
−0.02
(0.16)
0.25
(0.11)
−0.15
(0.17)
−0.26
(0.14)
3.60
(6.27)
−0.08
(0.20)
—
—
—
—
−0.26
(0.19)
−2.83
(0.63)
−0.01
(0.01)
0.01
(0.01)
−0.02
(0.01)
0.01
(0.01)
0.00
(0.01)
−10.14
(2.35)
0.01
(0.02)
—
—
0.00
(0.01)
—
—
−2.65
(0.50)
0.01
(0.01)
−0.01
(0.01)
0.01
(0.01)
0.02
(0.01)
−0.01
(0.01)
6.23
(1.67)
−0.01
(0.01)
—
—
0.01
(0.01)
—
—
1 Tick
Buy Orders
1 Tick
Sell Orders
BHO
21.70
(6.64)
−0.14
(0.11)
−0.29
(0.13)
−0.10
(0.10)
−0.02
(0.11)
−0.06
(0.08)
−11.32
(3.26)
—
—
−0.03
(0.14)
−0.13
(0.12)
—
—
Marg.
0.72
(1.22)
0.07
(0.04)
−0.06
(0.03)
0.05
(0.04)
−0.05
(0.04)
0.00
(0.03)
−13.70
(2.24)
0.12
(0.05)
—
—
−0.17
(0.04)
—
—
1 Tick
Sell Orders
ERR
10.64
(5.06)
0.21
(0.10)
−0.23
(0.09)
−0.01
(0.09)
−0.07
(0.10)
0.01
(0.06)
9.08
(3.85)
−0.12
(0.15)
—
—
—
—
0.23
(0.16)
Marg.
−1.21
(1.18)
0.03
(0.03)
0.00
(0.03)
−0.03
(0.04)
−0.02
(0.03)
−0.02
(0.02)
16.85
(2.30)
0.05
(0.04)
—
—
−0.06
(0.04)
—
—
1 Tick
Buy Orders
3.21
(4.79)
0.28
(0.13)
−0.10
(0.14)
0.07
(0.14)
0.18
(0.12)
0.07
(0.12)
−15.30
(4.63)
—
—
0.31
(0.14)
−0.30
(0.14)
—
—
Marg.
−6.78
(1.83)
0.15
(0.05)
−0.02
(0.03)
0.00
(0.03)
0.05
(0.03)
−0.06
(0.03)
−12.39
(3.24)
0.04
(0.04)
—
—
−0.12
(0.04)
—
—
1 Tick
Sell Orders
−11.83
(7.35)
0.10
(0.10)
−0.05
(0.14)
−0.11
(0.17)
−0.04
(0.12)
−0.27
(0.13)
16.58
(3.78)
0.03
(0.20)
—
—
—
—
−0.07
(0.17)
−9.24
(1.88)
0.03
(0.02)
−0.05
(0.03)
−0.03
(0.03)
0.01
(0.02)
−0.01
(0.03)
5.71
(2.97)
−0.02
(0.03)
—
—
0.01
(0.03)
—
—
1 Tick
Buy Orders
Marg.
WEM
Parameter estimates, with heteroscedasticity adjusted asymptotic standard errors in parentheses, for the regressions used to predict changes in
the common value conditional on order execution. The F-test is for the null that the state vector zt does not affect the conditional expectation, with
p-values in parentheses. The three sample stocks are Barkhor Resources (BHO), Eurus Resources (ERR), and War Eagle Mining Company (WEM).
Regression Model for Common Value Changes
2780
The Journal of Finance
R2
F-test
p-value
No. of Obs.
Sixth hour
Fifth hour
Fourth hour
Third hour
Second hour
Order
quantity
Recent
trades
Lagged
duration
Mid-quote
volatility
Distance to
midquote
First hour
0.01
(0.15)
0.00
(0.00)
−0.06
(0.03)
−2.34
(1.42)
−57.05
(6.72)
−0.62
(0.70)
0.20
(0.66)
−0.05
(0.66)
−0.63
(0.64)
−0.06
(0.66)
−0.56
(0.69)
0.33
7.64
(0.00)
293
−0.03
(0.01)
0.00
(0.01)
0.00
(0.00)
1.79
(0.34)
−7.50
(1.32)
−0.12
(0.04)
−0.03
(0.05)
0.03
(0.04)
−0.02
(0.04)
−0.00
(0.04)
0.01
(0.04)
0.13
12.45
(0.00)
2,388
0.21
(0.23)
0.02
(0.02)
0.16
(0.03)
1.53
(3.00)
−42.05
(12.23)
0.81
(1.01)
0.77
(1.01)
0.87
(0.98)
0.89
(1.10)
0.47
(1.02)
0.08
(1.04)
0.44
8.64
(0.00)
170
0.00
(0.02)
0.00
(0.01)
0.00
(0.00)
1.47
(0.30)
−5.31
(0.95)
−0.03
(0.04)
−0.01
(0.04)
0.04
(0.04)
0.01
(0.04)
0.01
(0.04)
0.06
(0.04)
0.09
6.90
(0.00)
2,261
−0.18
(0.13)
0.00
(0.03)
−0.03
(0.05)
−8.85
(3.78)
−128.71
(16.58)
−0.84
(0.60)
−0.72
(0.60)
−0.55
(0.57)
−0.59
(0.61)
−0.36
(0.59)
−0.44
(0.60)
0.28
5.84
(0.00)
565
0.00
(0.04)
−0.00
(0.01)
−0.02
(0.01)
−0.09
(0.71)
−33.59
(5.50)
−0.12
(0.16)
−0.31
(0.15)
0.01
(0.15)
0.17
(0.16)
−0.36
(0.15)
0.12
(0.15)
0.14
6.21
(0.00)
1,194
0.21
(0.13)
0.00
(0.03)
0.05
(0.04)
−9.08
(3.18)
−149.04
(18.28)
−0.21
(0.50)
−0.05
(0.50)
−0.51
(0.52)
−0.41
(0.51)
−0.41
(0.52)
−0.52
(0.69)
0.40
6.58
(0.00)
484
0.15
(0.04)
0.00
(0.01)
−0.01
(0.01)
0.62
(0.70)
−45.03
(5.59)
−0.26
(0.14)
−0.33
(0.13)
−0.10
(0.13)
−0.10
(0.12)
−0.30
(0.12)
−0.11
(0.13)
0.20
8.66
(0.00)
955
−0.17
(0.17)
−0.03
(0.02)
−0.03
(0.02)
0.10
(2.80)
−54.86
(14.63)
0.56
(0.47)
0.72
(0.47)
0.77
(0.42)
0.54
(0.44)
−0.31
(0.46)
0.20
(0.42)
0.35
6.77
(0.00)
327
−0.05
(0.05)
−0.00
(0.01)
−0.00
(0.01)
4.24
(1.07)
−20.18
(5.80)
−0.02
(0.16)
−0.08
(0.14)
−0.24
(0.13)
−0.21
(0.14)
−0.03
(0.14)
−0.26
(0.12)
0.20
4.08
(0.00)
892
0.19
(0.18)
0.07
(0.02)
0.02
(0.03)
4.85
(4.44)
−55.17
(22.80)
−1.17
(0.54)
−1.04
(0.55)
−0.63
(0.50)
0.15
(0.58)
0.07
(0.55)
−0.37
(0.64)
0.29
6.39
(0.00)
281
0.01
(0.03)
0.00
(0.01)
0.01
(0.01)
5.42
(1.11)
−11.24
(4.55)
−0.13
(0.12)
−0.06
(0.09)
−0.06
(0.09)
0.08
(0.07)
0.02
(0.08)
−0.03
(0.07)
0.09
3.45
(0.00)
858
Estimating the Gains from Trade
2781
2782
The Journal of Finance
The conditional probability of a buy market order between t and t + dt is the
probability that a trader who arrives finds it optimal to submit a buy market
order times the probability that a trader arrives. Using the conditional distribution of the private values, G(u | zt ), the trader arrival rate, and the assumption
that the one tick buy limit order is an optimal order submission for some trader,
we have
buy
Pr (Buy market order in [t, t + dt) | z t ) = Pr y t + ut ≥ θ0,1 (z t ) z t λ(t; xt ) dt
buy
= 1 − G θ0,1 (z t ) − y t xt λ(t; xt ) dt.
(50)
Similarly, the probability of a buy limit order is
Pr (Buy limit order in [t, t + dt) | z t )
buy
buy
= G θ0,1 (z t ) − y t xt − G θmarginal (z t ) − y t xt λ(t; xt ) dt.
(51)
The probability of a sell market order and a sell limit order are computed
similarly.
We may observe no orders submitted between t and t + dt for two reasons.
Either a trader does not arrive, or a trader arrives and chooses not to submit
any order, so that
Pr (No order submission in [t, t + dt) | z t )
buy
sell
= 1 − λ(t; xt ) dt + G θmarginal (z t ) − y t xt − G θmarginal
(z t ) − y t xt λ(t; xt ) dt.
(52)
The conditional probabilities in equations (50)–(52) depend on the trader
arrival rates, λ(t; xt ), the private values distribution, G(u | xt ), and the threshold
values, that is, the θ s. The threshold values themselves depend on the costs, the
execution probabilities, and the picking-off risks—see equations (16)–(20). We
substitute our first-step estimates of the execution probabilities and picking-off
risks into the threshold functions to form the log-likelihood function.
Let ti be the time of the ith order arrival. We use a Weibull parameterization
for the trader arrival rate. Suppose that the last order submission was at time
ti−1 . The arrival rate is
λ(t; xti ) dt = exp(xti δ)η(t − ti−1 )η−1 dt,
(53)
where xti denotes the exogenous state variables immediately before the order
submission at time ti . The private value distribution is parameterized as a
mixture of two normal distributions with standard deviations depending on
the common value and exogenous state variables,
u
u
G(u | xt ) = ρ
+
(1
−
ρ)
,
(54)
y t σ1 exp(xt )
y t σ2 exp(xt )
Estimating the Gains from Trade
2783
where denotes the standard normal cumulative distribution function, the
parameter satisfies σ1 = σ2 , and 0 < ρ < 1, and xt denotes the exogenous state
variables. Because the standard deviation is proportional to the common value,
the private values are normalized as percentages of the common value.
Table VII reports the conditional maximum likelihood estimates of parameters of the arrival rates and traders’ private values and execution costs, ce , with
standard errors reported in parentheses. The likelihood function is relatively
f lat with respect to the order submission cost, for positive order submission
costs. We therefore do not estimate the order submission cost, co , but set it
equal to 0.1% of the common value.
The first row of the top panel reports estimates of the Weibull shape parameter, η. The parameter estimate is less than one for all stocks: The longer the
time since the last order submission, the lower the conditional probability that
a new trader will arrive. The second through seventh rows of the top panel
report estimates of the parameters on the exogenous state variables. The exogenous state variables are standardized by dividing by their sample standard
deviations.
The estimated parameters on many of the exogenous state variables are positive. The arrival rate of traders increases following periods of high TSX market
volatility. The parameters on stock volatility are positive and larger in magnitude than the parameters on the other exogenous state variables. The relative
magnitudes of the parameters suggest that stock-specific shocks are more important than market-wide shocks for the trader arrival rates. Higher stock
volatility predicts an increase in the trader arrival rates.
The second panel reports parameter estimates for the distributions of the
private values. For all stocks, the private values distribution is a mixture of two
normal distributions, with approximately 85% weight on a distribution with a
standard deviation of approximately 5% and a 15% weight on a distribution
with a standard deviation of approximately 54%. The point estimates of the
order execution costs, ce , are reported in the third panel, and are between 1%
and 2% of the common value.
The estimated parameters on many of the exogenous state variables are statistically significant. A change in stock volatility has the largest effect on the
distributions of the private values. The parameters on stock volatility are all
negative; when stock volatility is high, a higher fraction of the traders that
arrive have private values close to zero. One interpretation of this relation is
persistence in stock volatility: If high stock volatility is often associated with
past high stock volatility and high volume, then perhaps the traders with the
most extreme private values have already altered their portfolios so that the
traders left have less extreme private values.
Our estimates assume that the exogenous variables follow a strictly Markov
process. Some of the results may change with the addition of additional lags.
We believe that our estimates of the gains from trade are likely to be robust to
the inclusion of additional lags of the exogenous state variables, however.
Table VIII reports the expected utilities for traders with six different private values across three different market conditions, namely, a low liquidity
The Journal of Finance
2784
Table VII
Conditional Maximum Likelihood Estimates
The table reports estimates of the trader arrival rate, the private value distribution and the trading
opportunity cost. Asymptotic standard errors are reported in parentheses. The table also reports
χ 2 tests for the null hypothesis of a constant trader arrival rate and a constant private value
variance, with p-values in parentheses. The three sample stocks are Barkhor Resources (BHO),
Eurus Resources (ERR), and War Eagle Mining Company (WEM).
BHO
ERR
WEM
Trader Arrival Rate
0.45
(0.00)
−2.15
(0.02)
0.03
(0.01)
0.03
(0.01)
−0.11
(0.01)
0.11
(0.01)
0.27
(0.01)
0.47
(0.00)
−2.08
(0.02)
0.02
(0.01)
−0.00
(0.01)
−0.00
(0.01)
0.06
(0.01)
0.15
(0.01)
0.45
(0.00)
−2.55
(0.02)
0.03
(0.01)
0.01
(0.01)
0.07
(0.01)
−0.04
(0.01)
0.19
(0.01)
Private Value Distribution
0.85
(0.04)
0.06
(0.00)
0.47
(0.04)
0.89
(0.03)
0.04
(0.00)
0.68
(0.06)
0.84
(0.05)
0.04
(0.00)
0.48
(0.05)
Time-Varying Variance
−0.00
(0.01)
−0.03
(0.01)
0.07
(0.01)
−0.04
(0.01)
−0.10
(0.01)
0.03
(0.01)
−0.02
(0.01)
−0.05
(0.01)
−0.02
(0.01)
−0.14
(0.01)
−0.05
(0.01)
−0.05
(0.01)
−0.03
(0.01)
0.03
(0.01)
−0.12
(0.01)
0.02
(0.00)
0.02
(0.00)
0.01
(0.00)
3871.95
(0.00)
434.55
(0.00)
680.94
(0.00)
814.08
(0.00)
985.28
(0.00)
369.70
(0.00)
Shape
Constant
TSX market volatility
TSX mining volatility
Interest rate volatility
Exchange rate volatility
Stock volatility
Mixing probability, ρ
σ1
σ2 − σ1
TSX market volatility
TSX mining volatility
Interest rate volatility
Exchange rate volatility
Stock volatility
Order execution cost, ce
χ 2 (5): Constant arrival rate
χ 2 (5): Constant private value variance
Table VIII
2.35
1.51
Sell 1 tick
Sell market
1.79
1.79
−3.69
−0.77
1.50
Buy market
Buy 1 tick
Buy marginal
2.50
0.44
1.63
4.13
Sell market
No order
−1.52
2.81
Sell 1 tick
Sell marginal
2.01
2.12
Sell marginal
0.77
−0.66
1.97
1.73
1.18
−1.61
−3.91
−0.03
−1.74
1.79
2.25
1.83
8.81
5.40
5.23
4.61
3.37
1.28
−11.70
1.15
−4.36
5.02
3.93
4.61
−5.08
4.66
4.16
4.61
−3.13
3.96
4.63
4.61
0.77
3.61
4.86
4.61
2.73
3.97
4.48
4.80
4.61
3.84
2.35
−6.86
1.55
4.10
4.50
4.61
4.07
2.89
−4.44
−3.29
3.35
3.91
4.61
4.54
3.97
0.40
−5.71
2.97
3.61
4.61
4.78
4.51
2.82
6.17
2.05
0.00
−4.12
−6.17
4.56
−0.28 −2.70
−7.54
−9.96
High liquidity state: Narrow spread, high depth
5.72
3.46
4.61
−8.99
−5.00
−2.50 −1.25
1.25
2.50
−9.68
−4.84 −2.42
2.44
4.84
Low liquidity state: Wide spread, low depth
−13.65
−8.81 −6.39
−1.55
0.87
−0.23
1.79
2.00
2.99
4.13
−4.33
−6.31
0.75
2.22
1.79
3.20
2.18
5.00
4.25
ERR
Private Value
−10.55
2.22
3.02
4.61
5.25
5.58
7.66
−10.29
−14.80
2.90
5.33
4.61
6.63
5.71
5.00
9.68
−2.40
−4.99
0.92
1.79
2.86
1.90
2.81
−3.99
−7.16
0.40
1.79
4.07
2.03
4.98
2.48
−0.87
6.22
4.61
3.72
4.24
−5.12
−1.08
−5.91
4.65
4.61
4.12
4.38
−0.08
−2.86
−8.43
3.87
4.61
4.32
4.45
2.44
−6.42
−13.47
2.30
4.61
4.72
4.58
7.48
−8.20
−15.99
1.52
4.61
4.92
4.65
10.00
2.84
2.36
2.53
−11.75
−21.03
1.54
4.61
−0.77
1.25
−6.04
6.27
3.94
2.28
1.54
1.11
−0.91
−7.23
3.64
3.56
2.67
0.97
1.54
−7.12
−9.94
−5.00
−6.75
−0.05
5.32
4.79
15.04
Moving market state: Common value proxy higher than the mid-quote
−2.24
−1.18
1.39
1.79
1.84
0.53
1.51
2.01
2.17
0.91
1.71
1.79
1.04
1.70
−0.44
0.95
1.76
1.67
1.79
1.79
2.05
1.77
1.77
0.95
1.18
−2.46
−4.19
1.40
2.00
1.79
1.32
0.06
2.50
2.12
−1.52
−3.13
1.88
1.79
0.38
−1.00
1.25
1.06
1.61
0.95
−1.18
0.35
−1.00
1.83
1.65
1.79
−1.51
−3.13
−1.25
−1.06
1.88
1.79
1.79
No order
1.54
0.55
1.38
−2.24
−4.36
−0.27
Buy marginal
Buy 1 tick
Buy marginal
1.28
0.06
2.04
2.47
3.16
2.18
Sell marginal
Sell 1 tick
Sell market
1.54
1.79
−2.45
−4.33
Buy 1 tick
1.31
1.79
−4.19
−6.31
Buy market
Buy marginal
No order
−2.50
−2.12
−5.00
−4.25
Percent
Cents
BHO
Private Value
0.99
−1.08
1.85
1.54
0.38
1.40
−2.60
2.89
2.48
1.85
1.54
1.28
0.11
−3.86
1.64
0.18
2.11
1.27
1.54
−4.26
−6.57
−2.50
−3.38
0.06
−2.81
1.51
1.54
0.96
1.47
−0.87
1.20
1.75
1.63
1.54
1.37
0.62
−2.17
0.64
−1.51
1.83
1.42
1.54
−2.84
−4.88
−1.25
−1.69
0.49
−2.71
−7.97
−1.78
−6.25
1.54
2.68
1.70
4.29
−3.86
−0.44
0.99
1.54
1.64
2.15
2.89
−2.36
−6.57
1.00
1.87
1.54
1.45
0.18
2.50
3.38
0.83
1.54
2.11
1.62
2.57
−2.17
0.29
1.20
1.54
1.55
1.64
1.20
−1.36
−4.88
1.28
1.72
1.54
0.02
−1.51
1.25
1.69
WEM
Private Value
−4.56
−11.41
−0.19
1.54
3.83
1.84
7.73
−7.23
−1.90
0.55
1.54
1.81
3.17
6.26
−4.36
−9.95
0.44
2.17
1.54
4.31
3.55
5.00
6.75
The table reports, for traders with different private values, the expected utilities for alternative order submissions. The expected utilities are computed for three different states, namely,
a low liquidity state in which the bid–ask spread is one standard deviation above its mean and the depth variables are one standard deviation below their means, a high liquidity state
in which the bid–ask spread is one standard deviation below its mean and the depth variables are one standard deviation above their mean values, and a moving market state in which
the common value proxy divided by the mid-quote is one standard deviation above its mean. All other variables are at their in-sample mean values. All utilities are reported in cents
per share. The maximum utility is indicated for each private value and market condition with a box. The three sample stocks are Barkhor Resources (BHO), Eurus Resources (ERR),
and War Eagle Mining Company (WEM).
Expected Utility by Trader Valuation
Estimating the Gains from Trade
2785
2786
The Journal of Finance
state with a wide spread and low depth, a high liquidity state with a narrow
spread and high depth, and a moving market state where the common value
is above the mid-quote. For each private value, we report the expected utility
from submitting a market order, a one-tick limit order, a marginal limit order,
or no order at all. The private values are 1.25%, 2.5%, or 5% higher or lower
than the common value; the corresponding private values measured in cents
are reported in the second row. The reported expected utility is a lower bound
on the trader’s expected utility because we do not compute the expected utility
for limit orders between one-tick from the quotes and marginal limit orders.
The maximum utility is indicated for each private value and market condition
with a box.
In the low liquidity state, a trader with a private value equal to 2.5% or −2.5%
optimally submits a marginal limit order, and a trader with private value equal
to 5% or −5% optimally submits a one-tick limit order. In the high liquidity state,
a trader with a private value equal to 1.25% or −1.25% optimally submits no
order in BHO and Eurus Resources (ERR), and optimally submits a limit order
in War Eagle Mining Company (WEM). A trader with a private value equal to
2.5% or −2.5% optimally submits limit orders for BHO and ERR and submits
market orders for WEM. A trader with a private value equal to 5% or −5%
optimally demands liquidity by submitting a market order for all stocks.
In the moving market state, the optimal order strategies are asymmetric.
With a high common value relative to the mid-quote, the optimal strategy is
to submit market orders in ERR and WEM for traders with all three positive
private values. Such market order submissions pick off some sell limit orders.
For BHO, traders with private values equal to 1.25% and 2.5% submit one-tick
limit orders rather than market orders.
The expected utilities reported in Table VIII show that traders’ optimal order
submission strategy is state dependent: For a given private value, the optimal
order submission changes with the state vector.
Table IX reports summary statistics for the estimated private value distributions and the optimal order submission strategies for five intervals for the
private value. The first row in each panel reports the mean proportion of traders
in each private value interval. The next rows report the mean and standard deviations of the fitted order submission probabilities for a sell market order, a
sell limit order, no order, a buy limit order, and a buy market order.
Traders with extreme private values typically choose market or limit orders
and rarely choose not to submit an order. Traders with intermediate private
values submit limit orders most frequently but also use market orders and
sometimes choose not to submit orders. Traders with private values close to
zero almost always submit limit orders when they choose to submit an order,
but often choose not to submit an order.
The standard deviations of the order submission probabilities reported in
Table IX indicate that for all five private value intervals, the state dependence
in the optimal order submission strategy is economically significant—traders’
optimal order submissions change frequently as the traders’ information set
and the limit-order book changes.
Estimating the Gains from Trade
2787
Table IX
Order Submission Probabilities by Trader Private Value
The table reports the in-sample mean of the probability of drawing a private value (u) from five different intervals.
For each interval and stock the table reports the in-sample mean and standard deviation of the probability of a
trader optimally submitting a sell market order, a sell limit order, no order, a buy limit order, or a buy market
order, conditional on the trader’s private value being in that interval. The three sample stocks are Barkhor
Resources (BHO), Eurus Resources (ERR), and War Eagle Mining Company (WEM).
Private Value
Order Submission
Mean probability of private
value in interval
Sell market
Sell limit
No order
Buy limit
Buy market
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean probability of private
value in interval
Sell market
Sell limit
No order
Buy limit
Buy market
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean probability of private
value in interval
Sell market
Sell limit
No order
Buy limit
Buy market
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
Mean
Standard deviation
(−∞, −5%]
(−2.5%, +2.5%)
[2.5%, +5%)
[ + 5%, +∞)
0.32
0.13
0.21
Order Submission Probabilities
0.74
0.25
0.01
0.30
0.36
0.07
0.24
0.64
0.25
0.29
0.39
0.20
0.02
0.09
0.36
0.09
0.25
0.28
0.00
0.02
0.37
0.03
0.13
0.27
0.00
0.00
0.01
0.00
0.01
0.05
0.00
0.01
0.00
0.04
0.07
0.22
0.74
0.36
0.19
0.32
0.00
0.00
0.00
0.01
0.01
0.07
0.30
0.31
0.69
0.31
0.48
0.14
0.12
Order Submission Probabilities
0.85
0.32
0.01
0.23
0.35
0.06
0.15
0.66
0.16
0.23
0.36
0.12
0.00
0.03
0.61
0.03
0.12
0.15
0.00
0.00
0.21
0.00
0.02
0.12
0.00
0.00
0.01
0.00
0.00
0.04
0.00
0.03
0.00
0.03
0.01
0.07
0.75
0.33
0.24
0.33
0.00
0.01
0.00
0.01
0.00
0.02
0.22
0.26
0.78
0.26
0.52
0.13
0.11
Order Submission Probabilities
0.92
0.48
0.04
0.16
0.41
0.09
0.08
0.51
0.41
0.16
0.41
0.17
0.00
0.01
0.12
0.01
0.05
0.18
0.00
0.00
0.40
0.00
0.02
0.12
0.00
0.00
0.03
0.00
0.01
0.09
0.00
0.00
0.00
0.05
0.01
0.09
0.62
0.42
0.37
0.41
0.00
0.00
0.00
0.01
0.00
0.02
0.16
0.24
0.84
0.24
0.21
0.12
0.11
(−5%, −2.5%]
BHO
0.13
ERR
0.14
WEM
0.13
The Journal of Finance
2788
IV. Estimates of the Gains from Trade
Substituting the mixture of normal distributions assumption for the private
values distribution in equation (54) into the maximum gains from trade in
equation (33), the maximum gains from trade as a percentage of the common
value is
Maximum gains (xti )
= ρ 2σ1 exp xti φ
c e + co
−ce − co
− (ce + co )2
σ1 exp xti exp σ1 xti c e + co
−ce − co
− (ce + co )2
+ (1 − ρ) 2σ2 exp xti φ
.
σ2 exp xti σ2 exp xti (55)
Here φ is the standard normal density function, and is the standard normal
cumulative distribution function. In Appendix E we derive equation (55) and
provide a description of how we compute the current gains, and the losses from
no execution, no submission, wrong direction, extramarginal submission, and
the monopoly gains.
The current gains from trade in the limit-order market depend on the threshold valuations and the execution probabilities for limit orders. We estimate
the conditional execution probabilities for the marginal and the one-tick buy
and sell limit orders. If in any given state the execution probabilities for any
buy limit order are greater than or equal to the execution probability for the
marginal buy limit order, then using the execution probability for the marginal
buy limit order for all buy limit orders in that state provides a lower bound on
the current gains from trade on the buy side. An analogous argument applies
to the sell side. Similarly, using the execution probabilities for one-tick limit
orders for all limit order execution probabilities provides an upper bound on
the gains from trade. We report the lower and upper bounds for the current
gains from trade. We also report the average current gains from trade, that is,
the gains computed using the average of the one-tick and marginal buy or sell
execution probabilities for all buy or sell limit orders in a given state.
We take expectations across states zti and report the unconditional expected
current and maximum gains from trade. For brevity we refer to the expected
gains simply as the current gains from trade.
We observe an order submission when a trader arrives and submits an order.
Suppose an order is submitted at time ti with the state vector equal to zti . We
compute the expected gains from trade at that information set, allowing us
to obtain a sample of expected gains sampled at every order submission. Let
Current gains(z ti ) be the resulting estimate of the current gains. Some traders
who arrive may find it optimal to submit no order, implying that our sample is
drawn conditional on observing an order submission.
Let I be the total number of observations in our sample. Because we only
compute gains when we observe an order submission, the sample average
Estimating the Gains from Trade
2789
I
1
Current gains(z ti )
I i=1
(56)
E[Current gains(z t ) | Trader arrives and submits and order].
(57)
is an estimate of
But this measure misses the gains from trade when a trader arrives at the
market but chooses not to submit an order. To include such cases, we wish to
measure the gains from trade conditional on a trader arriving,
E[Current gains(z t ) | Trader arrives].
(58)
We reweight our sample to estimate equation (58). In Appendix F we show
that
Pr (z t | Trader arrives) = Pr (z t | Trader arrives and submits an order) × ω(z t ),
(59)
with
ω(z t ) =
Pr (Trader submits an order | Trader arrives)
.
Pr (Trader submits an order | z t , Trader arrives)
(60)
From equations (59) and (60), we have
E[Current gains(z t ) × ω(z t ) | Trader arrives and submits and order]
= E[Current gains(z t ) | Trader arrives].
(61)
We describe how we use our first-stage and maximum likelihood estimates to
obtain consistent estimates of ω(zti ) in Appendix F. We use the sample average
I
1
Current gains(z ti ) × ω̂(z ti )
I i=1
(62)
to form a consistent estimate of E[Current gains(zt ) | Trader arrives]. Our estimates of the maximum gains from trade, the losses from no execution, no
submission, wrong direction, and extramarginal submission, and the monopoly
gains are all weighted in the same way.
The top panel of Table X reports estimates of the maximum and the current
gains from trade. The gains are reported as a percentage of the common value.
The first row reports the maximum gains from trade. In the next three rows
we report estimates of the lower and upper bounds for the current gains from
trade and the average current gains from trade. In rows five through seven we
report the difference between the maximum and the current gains from trade
and in the last three rows of the top panel we report the current gains from
trade as a percentage of the maximum gains from trade.
Across the three stocks, the average of the maximum gains is approximately
8.2% of the common values and the average of the lower bound on the current
2790
The Journal of Finance
Table X
Estimates of the Gains from Trade
The first row of the top panel reports the estimates of the maximum gains from trade, measured as a
percent of the common value. The next three rows report the estimates of the lower and upper bounds
and the average current gains from trade; details are provided in Appendix E. The next six rows report
the lower and upper bounds and the average for the difference between the maximum and the current
gains from trade, and the current gains from trade as a percentage of the maximum gains from trade.
The middle panel reports the average percentage of the efficiency loss associated with no execution, no
submission, wrong direction, and extramarginal submissions computed for the average current gains
from trade. The bottom panel reports the monopoly gains from trade as a percentage of the common
value, and the current gains as a percentage of the monopoly gains. We verify that the second-order
conditions hold at the computed monopoly bid and ask quotes at each observation. The three sample
stocks are Barkhor Resources (BHO), Eurus Resources (ERR), and War Eagle Mining Company (WEM).
BHO
Lower bound
Upper bound
Average
Lower bound
Upper bound
Average
Lower bound
Upper bound
Average
Sell side
Buy side
Subtotal
Sell side
Buy side
Subtotal
Sell side
Buy side
Subtotal
Sell side
Buy side
Subtotal
Total
ERR
Gains
Maximum gains as a % of the common value
9.07
8.61
Current gains as a % of the common value
7.88
8.09
8.45
8.31
8.16
8.20
Maximum gains minus current gains
0.62
0.30
1.20
0.52
0.91
0.41
Current gains as a % of maximum gains
86.79
93.97
93.13
96.57
89.96
95.27
Decomposition of Losses
No execution as a % of total losses
32.32
31.20
40.10
39.01
72.42
70.21
No submission as a % of total losses
2.24
0.62
1.98
0.15
4.22
0.77
Wrong direction as a % of total losses
0.86
0.02
0.20
0.05
1.06
0.07
Extramarginal submissions as a % of total losses
9.81
11.87
12.49
17.07
22.30
28.94
100.00
100.00
Monopoly Gains
Monopoly gains as a % of the common value
5.02
5.57
Monopoly gains as a % of maximum gains
55.31
64.71
Current gains as a % of monopoly gains
162.65
147.23
WEM
6.75
6.08
6.40
6.24
0.35
0.67
0.51
90.07
94.81
92.44
33.05
41.85
74.90
0.41
0.71
1.12
0.39
0.63
1.02
10.30
12.66
22.96
100.00
4.18
61.87
149.41
Estimating the Gains from Trade
2791
gains is around 7.4% of the common values. The maximum benefit of increasing the efficiency of the allocations resulting from current market design—the
limit-order market—is less than 1% of the stocks’ common values. The current
gains from trade are approximately 90% of the maximum gains from trade.
We interpret the magnitudes as evidence that in our sample, the limit-order
market allows the traders to achieve a relatively efficient stock allocation.
From Table I, approximately 60% of the order submissions in our sample
are limit orders. The average fitted execution probability for limit orders is
approximately 40% across the three stocks, implying that approximately 36%
of the order submissions result in no execution. If we ignore the valuations of
the traders, the order submissions and fitted execution probabilities imply that
the current gains from trade are approximately 64% of the maximum gains from
trade. But we estimate the current gains from trade to be approximately 90%
of the maximum gains from trade. We estimate the current gains from trade
higher than 64% because the traders who contribute the most to the gains
from trade endogenously submit market orders and thereby avoid the risk of
no execution altogether, whereas the traders who contribute less to the gains
from trade endogenously bear most of the risk of no execution. For example,
from Table IX, approximately 80% of the traders with valuations more than 5%
away from the common value submit market orders, while approximately 65%
of the traders with valuations between 2.5% and 5% away from the common
value submit limit orders.
In our model, losses do not arise only from unexecuted order submissions.
Losses also arise from no submission, wrong direction, and extramarginal submission. The bottom panel of the table reports the decomposition of the average
losses as percentages of the total losses. No execution is the most important
sources of losses, accounting for between 70% and 75% of the losses.
No submission accounts for between 1% and 4% of the losses. Using the order
submission cost of 0.1% of the common value and the order execution cost of
approximately 2%, all traders with private values farther than 2.1% from the
common value should submit orders that execute. From Table IX, less than 3%
of the traders with valuations greater than 2.5% away from the common value
do not submit an order.
Wrong direction accounts for no more than 1.1% of the losses. The result
suggests that it is rare to see extremely stale limit orders that entice high
private value traders to sell or low private value traders to buy.
Extramarginal submission accounts for between 22% and 29% of the losses.
Using the order submission cost of 0.1% of the common value and the trading cost of approximately 2%, all traders with private values less than 2.1%
away from the common value should not submit orders. From Table IX, approximately 64% of the traders with valuations less than 2.5% away from the common
value submit an order. The losses from extramarginal submission are smaller
than 64% for two reasons. First, the cut-off of 2.5% includes some traders who
should trade. Second, all traders who submit orders that lead to losses from
extramarginal submissions have valuations that are less extreme than the valuations of traders who should trade.
2792
The Journal of Finance
The traders in our model solve a one-shot order submission problem and do
not have an opportunity to return to the market. In reality, however, some of
the order submissions that lead to wrong direction or extramarginal submission
may be reversed. Such trades that are later reversed may provide liquidity in a
speculative fashion and thereby improve efficiency. Our estimates of the current
gains may therefore be biased downward. Because wrong direction trades only
account for 1% of the losses and extramarginal submissions account for about
25% of the losses, the downward bias may be small, however. To the extent
our estimates are downward biased, then our estimates that the current limitorder market leads to the traders realizing 90% of the maximum gains is a
lower bound.
The bottom panel of Table X reports estimates of the monopoly gains from
trade. Across the three stocks, the average of the monopolist gains is approximately 5% of the common value, and 60% of the maximum gains. The monopolist
maximizes profits by charging a wide spread, which reduces trading relative to
the trading that leads to the allocation that maximizes the gains from trade.
We find that the current gains are approximately 150% of the monopoly gains.
Thus, the current limit-order system leads to a much more efficient allocation of
the stock than the monopolist generates. Most of the losses in the current limitorder market relative to the gains in the efficient allocation arise from nonexecution. Although the monopolist guarantees execution at his posted quotes,
his profit-maximizing spread is large enough to significantly reduce the trading relative to trading that leads to the allocation in the current limit-order
market. As a consequence, the monopoly gains are significantly lower than the
gains in the current limit-order market.
A critical assumption for the counterfactual calculations of the maximum
gains and the monopoly gains is that that the underlying arrival rate of traders,
λ, as well as the distribution of the traders’ private valuations, G, are invariant
to the changes in the trading rules. If changes in trading rules have feedback
effects on the arrival rate or the distribution of private valuations then the
resulting gains from trade are also likely to change. Notwithstanding such
issues, our gains calculations provide a good starting point for evaluating the
design of the limit-order market.
Overall, our empirical work provides evidence that the limit-order market allows traders to realize many of the potential gains from trade, and significantly
more gains than would be achieved under a profit-maximizing monopolist that
provides bid and ask quotes.
V. Conclusions
In a perfectly liquid market in which the traders can realize the maximum
gains from trade, the market’s design provides incentives to traders with high
private values to buy the stock, traders with low private values to sell the stock,
and traders with intermediate private values—traders who have no strong need
to trade—to abstain from trading. We compute the expected current, maximum,
and monopolist gains from trade using a sample of both order submissions and
execution and cancellation histories from a limit-order market. The current
Estimating the Gains from Trade
2793
gains are approximately 90% of the maximum gains from trade and 150% of
the monopolist gains from trade. The limit-order market therefore allows the
traders to realize many of the gains from trade.
Close to three-quarters of the difference between the current and maximum
gains from trade arise from nonexecution. Nonexecution arises because the
limit-order market provides incentives for traders to submit orders with high
execution risk. Almost one quarter of the difference between the current and
maximum gains comes from extramarginal submissions. Extramarginal submissions occur when traders with no strong need to trade find profitable trading
opportunities.
A key feature of our model is that we base our computations of the gains
from trade on estimates of the traders’ optimal order submission strategy in
the limit-order market. Our estimates condition on the endogeneity of traders’
order submissions. Such endogeneity is consistent with empirical evidence that
the composition of the order f low changes as the limit-order book and market
conditions change. Traders with more extreme private values tend to submit
orders with low execution risk and low picking-off risk and traders with more
moderate private values tend to submit limit orders with higher execution risk
and higher picking-off risk. Our estimates are also consistent with the ability
of traders to switch between market orders and limit orders when the relative payoffs of market and limit orders change. Both effects are important
empirically.
Because our empirical approach builds directly on the theoretical model, our
parameter estimates can be used in numerical solutions for the equilibrium and
allow researchers to study efficiency under alternative trading rules. Goettler,
Parlour, and Rajan (2005) use numerical methods to solve for the equilibrium of
a model similar to ours. They determine the impact of a change in the tick size
for a model using a normal distribution for the private values with a standard
deviation of $ 14 and zero costs. They report the current gains as $0.1693 in the
1
$ 18 -tick regime and $0.1728 in the $ 16
-tick regime. Plugging these figures into
equation (55), the maximum gains from trade are $0.1995; the current gains
1
are 84.9% of the maximum gains in the $ 18 -tick regime and 86.6% in the $ 16
-tick
regime.
Our model is based on a one-shot order submission problem with no endogenous order cancellations or order resubmissions. Much work on the efficiency of
double auctions argues that double auctions are relatively efficient because the
traders can continue to resubmit bids until many of the gains from trade are
exhausted. It would therefore be interesting to extend our model and empirical
approach to allow for the possibility of endogenous cancellations and resubmissions. The theoretical model in Rosu (2004) features endogenous cancellations
and resubmissions in a somewhat different setting from ours. His model could
provide a starting point for such empirical work. Allowing for endogenous cancellations and order resubmissions may increase our estimates of the efficiency
of the limit-order market.
Our model also takes as exogenous the order submission quantity. However, a trader is likely to choose how many shares to buy or sell depending on
his marginal valuations for additional shares. Studying the robustness of our
2794
The Journal of Finance
empirical findings to quantity dependent valuations and endogenous quantity
choice would be a useful direction for future research.
Some limit-order markets have added designated market makers who have
an obligation to submit limit orders in less actively traded stocks. The designated market makers typically pay lower order submission fees than other
traders, but the market makers face the same rules as all other traders submitting orders. For example, Euronext Paris has designated market makers for
small and less actively traded companies (Mann, Venkataraman, and Waisburd
(2002)). In order to evaluate whether adding the designated market makers
leads to a net efficiency improvement of the limit-order market, we need to
be able to measure the gains from trade with and without designated market
makers. Our method is well suited to generating such measurements.
Appendix A: Proofs
Proof of Lemma 1: The objective function in equation (29) is increasing in
the private value u for the traders who buy the stock in the optimal allocation:
If a trader with private value u buys the stock in the optimal allocation, then a
trader with valuation u > u also buys the stock in the optimal allocation. A similar result holds for traders who sell the stock in the optimal allocation. There
therefore exist two numbers, L(xt ) and H(xt ), such that the optimal allocation
is
1, ut ≤ L(xt ),
1, ut ≥ H(xt ),
sell∗
buy∗
I
, I
. (A1)
(ut ; xt ) =
(ut ; xt ) =
0, otherwise
0, otherwise
Let P(xt ) be a solution for P to
G(P − ce − co | xt ) − (1 − G(P + ce + co | xt )) = 0.
(A2)
As P goes to −∞, G(P − ce − co | xt ) goes to zero and (1 − G(P + ce + co | xt )) goes
to one; and as P goes to ∞, G(P − ce − co | xt ) goes to one and (1 − G(P + ce +
co | xt )) goes to zero. The function in equation (A2) is monotonic in P and by
the continuity of the distribution, G, the function is also continuous in P. A
unique solution therefore always exists to equation (A2). If the distribution is
symmetric, P(xt ) is the median of the distribution.
We now show that the cut-offs
L(xt ) = P (xt ) − ce − co , H(xt ) = P (xt ) + ce + co
(A3)
characterize the optimal allocation. By construction, the market clearing condition is satisfied at the allocation. Suppose that L̂ > P (xt ) − ce − co and Ĥ
characterize an alternative allocation. The market clearing condition for the
alternative allocation is
G( L̂ | xt ) = 1 − G( Ĥ | xt ).
(A4)
Estimating the Gains from Trade
2795
Equations (A2) and (A4) imply Ĥ < P (xt ) + ce + co and
L̂
P (xt )+ce +co
g (u | xt ) du −
g (u | xt ) du = 0.
P (xt )−ce −co
(A5)
Ĥ
Let I(ut ≤ L) be an indicator function for ut ≤ L, with I(ut ≥ H) defined similarly.
The difference in the expected gains from trade between the two allocations is
E I (ut ≤ L̂)(−ut − ce − co ) + I (ut ≥ Ĥ)(ut − ce − co ) | xt ]
− E[I (ut ≤ P (xt ) − ce − co )(−ut − ce − co )
+ I (ut ≥ P (xt ) + ce + co )(ut − ce − co ) | xt ]
L̂
P (xt )+ce +co
=
(−u − ce − co ) g (u | xt ) du +
(u − ce − co ) g (u | xt ) du
<
P (xt )−ce −co
L̂
P (xt )−ce −co
+
Ĥ
(−(P (xt ) − ce − co ) − ce − co ) g (u | xt ) du
P (xt )+ce +co
((P (xt ) + ce + co ) − ce − co ) g (u | xt ) du
Ĥ
= P (xt ) −
L̂
P (xt )−ce −co
g (u | xt ) du +
P (xt )+ce +co
g (u | xt ) du
Ĥ
= 0,
(A6)
where the inequality follows from the monotonicity in u of both integrals.
Similar logic holds for alternative allocations with L̂ < P (xt ) − ce −
co . Q.E.D.
Proof of Lemma 2: Using the logic in the proof to Lemma 1, the trader’s
optimal decision can still be characterized by two cut-offs. A trader optimally
buys if the utility from buying is greater than the utility from not trading, that
is,
y t + ut − ce − co − atm ≥ 0,
(A7)
ut ≥ atm + ce + co − y t .
(A8)
implying the cut-off
An analogous result holds for the sell decision.
Using the distribution of private values, the monopolist’s profit-maximization
problem is
max
G btm − ce − co − y t xt y t − btm + 1 − G atm + ce + co − y t xt atm − y t ,
m, m
{bt ,at }
(A9)
subject to the market clearing condition
G btm − ce − co − y t xt = 1 − G atm + ce + co − y t xt .
(A10)
The Journal of Finance
2796
To solve the problem, we solve the monopolist’s problem without the market clearing constraint and show that the unconstrained problem satisfies the
constraint. The first-order condition for the bid quote bm∗
t in the unconstrained
problem is
g bm∗ − ce − co − y t xt y t − bm∗ − G bm∗ − ce − co − y t xt = 0,
(A11)
t
t
t
which can be rewritten as
m∗
b
G btm∗ − ce − co − y t xt
,
= y t − m∗
g bt − ce − co − y t xt
and the first-order condition for the ask quote can be rewritten as
1 − G atm∗ + ce + co − y t xt
m∗
.
a = yt +
g atm∗ + ce + co − y t xt
(A12)
(A13)
By the symmetry of the distribution and the zero median, the allocation at
these solutions satisfies market clearing. The second-order condition for the
bid quote is
m∗
g btm∗ − ce − co − y t xt y t − btm∗ − 2 g bt,0
− ce − co − y t xt ≤ 0,
(A14)
and the second-order condition for the ask quote is
m∗
g atm∗ + ce + co − y t xt atm∗ − y t + 2 g at,0
+ ce + co − y t xt ≤ 0.
(A15)
In order to show that the ask must be higher than the bid at the optimum,
substitute the constraint into the objective function to get
max
G btm − ce − co − y t xt atm − btm ,
(A16)
m, m
{bt ,at }
m
which is only positive if am
t > bt . The expression for the monopoly gains follows
from substituting the monopolist’s bid and ask quotes into the traders’ optimal
order submissions and taking conditional expectations. Q.E.D.
Appendix B: The Competing Risks Model
Here, we brief ly describe the competing risks model. Our discussion is based
on Kalbf leisch and Prentice (2002, Chapter 8) and Lancaster (1990, Chapter
5). Suppose that there are M possible failure types for a system and let Tm for
m = 1, 2, . . . , M denote the associated random failure times. We observe the
time at which the failure occurs, T, defined to be equal to the minimum of the
random failure times,
T = min(T1 , T2 , . . . , TM ),
th
(B1)
and the failure type. Here, we refer to Tm as the m latent time.
In our application, there are two reasons for an order to leave the limit-order
book, namely, a cancellation or an execution so M = 2. We observe the time
at which the order leaves the book and how the order leaves the book, either
through cancellation or through execution.
Estimating the Gains from Trade
2797
The hazard rate for Tm is defined as
Pr (Tm ∈ [τ, τ + τ ) | T ≥ τ, z t )
,
τ ↓0
τ
hm (τ ; Q) = lim
(B2)
where Q is a vector of conditioning variables. For an independent competing
risks model in which the Tm s are independent,
hm (τ ; Q) =
f m (τ | Q)
,
1 − Fm (τ | Q)
(B3)
where Fm (τ | Q) is the distribution of the mth latent time and fm (τ | Q) is the
density of the mth latent time.
Irrespective of whether the model is an independent competing risks model
or not, the hazard rate for T is
h(τ ; Q) =
M
hm (τ ; Q).
(B4)
m=1
The distribution of T is
F (τ | Q) = 1 − e−
τ
0
h(s;Q)ds
,
(B5)
with density
f (τ | Q) = h(τ ; Q)e−
τ
0
h(s;Q) ds
.
(B6)
The density of the mth time is
f m (τ | Q) = hm (τ ; Q)e−
τ
0
h(s;Q) ds
.
(B7)
Suppose we have a sample [τj , δj,1 , δj,2 , . . . , δj,M , Qj ] for j = 1, . . . , J, where τ j
is the time of the jth failure, δ j,m is an indicator for the jth failure type, and Qj is
the conditioning information associated with the jth failure. Using the densities
of the times, the likelihood function associated with these data is
!
J
M
τ
δt j ,m − 0 j h(s;Q j )d s
.
(B8)
hm (τ j ; Q j )
e
j =1
m=1
Taking logarithms and using the definition of the hazard rate for T, the loglikelihood function is
!
τj
J
M
(δ j ,m ln hm (τ j ; Q j )) −
h(s; Q j ) ds
j =1
=
J
M
j =1
=
0
m=1
j =1
τj
(δ j ,m ln hm (τ j ; Q j )) −
0
m=1
M
J
m=1
m=1
τj
(δ j ,m ln hm (τ j ; Q j )) −
0
M
!
hm (s; Q j ) ds
!
hm (s; Q j ) ds .
(B9)
The Journal of Finance
2798
In estimating the parameters of distributions of the time to cancellation and
the time to execution, we assume that execution times and cancellation times
follow independent Weibull distributions, with hazard rates as in equation (47)
in the text. Parameter estimates are found by maximizing the log-likelihood
function.
Appendix C: Computing Execution Probabilities
buy
Suppose a buy limit order at price pt,b is submitted at time t. Assume that
the latent cancellation and execution times have the distributions given by
equations (46) and (48) in the text. The execution probability is
buy
buy
ψb (z t ) = E It (τexecute ≤ τcancel ) z t , d t,b = 1
T
buy
buy
=
1 − Fcancel τ z t , d t,b = 1 dF execute τ z t , d t,b = 1
0
=
buy buy buy
buy buy exp −exp z t γb τ αb exp z t κb βb τ βb −1
0
buy buy × exp −exp z t κb τ βb −1 d τ.
T
(C1)
The second line follows from the independence of the latent cancellation and execution times, and the assumption that the latent cancellation time is bounded
by t + T with probability one. We compute equation (C1) numerically with T
equal to two trading days, or 48,600 seconds.
Appendix D: The Conditional Likelihood Function
We observe the time and type of each order submission and the state vector
at the time of each order submission. Let ti denote the time of the ith order
buy
and d ti ,b the decision
submission, I the total number of order submissions, d tsell
i ,s
indicators, and zti the state vector. We use these data to form a competing risks
model for the order submission times. Conditioning on the common value, order
quantity, xt and zt , using the traders’ decision rule and the trader arrival rate,
the hazard rate for a sell market order is
sell
G θ0,1
(z t ) − y t xt λ(t; xt ),
(D1)
the hazard rate for a sell limit order is
sell
sell
G θmarginal (z t ) − y t xt − G θ0,1
(z t ) − y t xt λ(t; xt ),
(D2)
the hazard rate for a buy market order is
buy
1 − G θ0,1 (z t ) − y t xt λ(t; xt ),
(D3)
and the hazard rate for a buy limit order is
buy
buy
G θ0,1 (z t ) − y t xt − G θmarginal (z t ) − y t xt λ(t; xt ).
(D4)
Estimating the Gains from Trade
2799
The hazard rate for an order submission of any type is the sum of the hazards for each different type of order submission in equations (D1)–(D4), and
equals
buy
sell
1 − G θmarginal (z t ) − y t xt + G θmarginal
(z t ) − y t xt λ(t; xt ).
(D5)
Substituting the hazard rates in equations (D1)–(D5) into the log-likelihood
function for a competing risks model in equation (B9), the conditional loglikelihood for the order submission times is:
I
d sell ln G θ sell (z t ) − y t xt λ(ti ; xt )
ti ,0
i=1
+
0,1
S(z
t )
d tsell
i ,s
i
i
i
i
sell
sell
ln G θmarginal
(z ti ) − y ti xti − G θ0,1
(z ti ) − y ti xti λ(ti ; xti )
s=1
buy
ln 1 − G θ0,1 (z ti ) − y ti xti λ(ti ; xti )
B(z
t ) buy
buy
buy
+
d ti ,b ln G θ0,1 (z ti ) − y ti xti − G θmarginal (z ti ) − y ti xti λ(ti ; xti )
buy
+ d ti ,0
−
b=1
ti
ti−1
1−G
buy
θmarginal (z t )
!
sell
− y t xt + G θmarginal (z t ) − y t xt λ(t; xt ) dt .
(D6)
In our estimation, we assume that the common value, yt , and state vector, zt ,
only change when an order is submitted. Using this assumption, we have
ti
buy
sell
1 − G θmarginal (z t ) − y t xt + G θmarginal
(z t ) − y t xt λ(t; xt ) dt
ti−1
buy
sell
= 1 − G θmarginal (z ti ) − y ti xti + G θmarginal
(z ti ) − y ti xti
ti
λ(t; xti ) dt.
ti−1
(D7)
Appendix E: Formulas for the Gains to Trade
Suppose that the random variable e is distributed as a mixture of normals,
with weight ρ 1 on a normal distribution with mean zero and standard deviation
σ 1 and weight 1 − ρ 1 on a normal distribution with mean zero and standard
deviation σ 2 . Let a ≤ b be two constants, and I(a ≤ e ≤ b) be an indicator
function. Then,
b
a
E[I (a ≤ e ≤ b)e] = ρ1 σ1 φ
− σ1 φ
σ1
σ1
b
a
(E1)
+ (1 − ρ1 ) σ2 φ
− σ2 φ
,
σ2
σ2
The Journal of Finance
2800
and
b
a
E[I (a ≤ e ≤ b)] = ρ1 −
σ1
σ1
a
b
+ (1 − ρ1 ) −
,
σ2
σ2
(E2)
where φ is the standard normal density function and is the standard normal
cumulative distribution function.
The private value is distributed as a mixture of normals, with standard deviations σ1 (xt ) = yt σ1 exp(xt ) and σ2 (xt ) = yt σ2 exp(xt ). The formula for the maximum gains from trade in equation (55) in the text results from applying equations (E1) and (E2) to equations (32) and (33), using the parameterization that
the order execution costs and the order submission costs are both proportional
to the common value: Specifically, order execution cost = yt ce and order submission cost = yt co .
In order to compute the current gains from trade and the losses from no execution, no submission, and extramarginal submissions, we need to compute
expectations over different intervals defined by combinations of threshold valuations and the cut-offs that define the optimal allocation.
Substituting the optimal order submission strategy for the sell side in equations (22)–(25) and the corresponding equations for the buy side into equation
(27), the current gains are
Current gains (z t )
sell
= E I ∞ ≤ y t + ut ≤ θ0,1
(z t ) (−ut − ce − co ) z t
+
S(z
t )−1
sell
sell
E I θi−1,i
(z t ) ≤ y t + ut ≤ θi,i+1
(z t ) ψisell (z t )(−ut − ce ) − co z t
i=1
sell
sell
sell
+ E I θ S(z
(z t ) ≤ y t + ut ≤ θmarginal
(z t ) ψ S(z
(z t )(−ut − ce ) − co z t
t )−1, S(z t )
t)
buy
+ E I θ0,1 (z t ) ≤ y t + ut ≤ ∞ (ut − ce − co ) z t
+
B(z
t )−1
buy
buy
buy
E I θi,i+1 (z t ) ≤ y t + ut ≤ θi−1,i (z t ) ψi (z t )(ut − ce ) − co z t
i=1
buy
buy
buy
+ E I θ B(z t )−1, B(z t ) (z t ) ≤ y t + ut ≤ θmarginal ψ B(z t ) (z t )(ut − ce ) − co z t .
(E3)
The lower bound for the current gains from trade is obtained by using the
execution probabilities for the marginal sell limit orders for all the sell limit
order execution probabilities and the execution probabilities for the marginal
buy limit order for all buy limit order execution probabilities:
Estimating the Gains from Trade
2801
sell
Current gains (z t ) ≥ E I ∞ ≤ y t + ut ≤ θ0,1
(z t ) (−ut − ce − co ) z t
sell
sell
sell
+ E I θ0,1
(z t ) ≤ y t + ut ≤ θmarginal
(z t ) ψ S(z
(z t )(−ut − ce ) − co z t
t)
buy
+ E I θ0,1 (z t ) ≤ y t + ut ≤ ∞ (ut − ce − co ) z t
buy
buy
buy
+ E I θmarginal ≤ y t + ut ≤ θ0,1 (z t ) ψ B(z t ) (z t )(ut − ce ) − co z t .
(E4)
The upper bound is obtained by using the execution probabilities for one-tick
sell and buy limit orders for all sell and buy limit order execution probabilities in
equation (E3). The average gains from trade are obtained by using the average
execution probabilities for the one-tick buy and marginal buy limit orders for
all buy limit order execution probabilities in equation (E3). The sell limit order
executions are defined similarly.
Using equations (E1) and (E2), we compute a closed form for each term in
equation (E4). For example,
sell
sell
sell
E I θ0,1
(z t ) ≤ y t + ut ≤ θmarginal
(z t ) ψ S(z
(z t )(−ut − ce ) − co z t
t)
sell
θmarginal (z t ) − y t
sell
= y t ψ S(z t ) (z t ) −ρ σ1 exp(xt )φ
y t σ1 exp(xt )
sell
(z
)
−
y
θ
t
t
0,1
− σ1 exp(xt )φ
y t σ1 exp(xt )
sell
sell
θmarginal (z t ) − y t
θ0,1 (z t ) − y t
− (1 − ρ) σ2 exp(xt )φ
− σ2 exp(xt )φ
y t σ2 exp(xt )
y t σ2 exp(xt )
sell
sell
θmarginal (z t ) − y t
θ0,1 (z t ) − y t
− ce ρ −
y t σ1 exp(xt )
y t σ1 exp(xt )
sell
sell
!
θmarginal (z t ) − y t
θ0,1 (z t ) − y t
+ (1 − ρ) −
y t σ2 exp(xt )
y t σ2 exp(xt )
sell
sell
θmarginal (z t ) − y t
θ0,1 (z t ) − y t
− y t co ρ −
y t σ1 exp(xt )
y t σ1 exp(xt )
sell
sell
!
θmarginal (z t ) − y t
θ0,1 (z t ) − y t
(E5)
+ (1 − ρ) −
.
y t σ2 exp(xt )
y t σ2 exp(xt )
The other terms and the upper bound are computed similarly.
The no execution losses, no submission losses, wrong direction losses, extramarginal submission losses, and the monopoly gains are computed similarly.
The Journal of Finance
2802
Appendix F: Reweighting the Sample
Using the definition of conditional probability,
Pr (z t | Trader arrives)Pr (Trader submits an order | z t , Trader arrives)
= Pr (z t , Trader submits an order | Trader arrives),
(F1)
or
Pr (z t | Trader arrives) =
Pr (z t , Trader submits an order | Trader arrives)
.
Pr (Trader submits an order | z t , Trader arrives)
(F2)
Also from the definition of conditional probability,
Pr (z t , Trader submits an order | Trader arrives)
= Pr (z t | Trader arrives and submits an order)
× Pr (Trader submits an order | Trader arrives).
(F3)
Substituting equation (F3) into equation (F2) and rearranging gives equations (59) and (60) in the text.
We use our first-stage and maximum likelihood estimators to estimate ω(zti ),
which from equation (60) depends on
Pr (Trader submits an order | Trader arrives)
and
Pr Trader submits an order | z ti , Trader arrives .
From the theoretical model,
Pr (Trader submits an order | z t , Trader arrives)
sell
buy
= G θmarginal
(z t ) − y t xt + 1 − G θmarginal (z t ) − y t xt .
(F4)
We use equation (F4) evaluated at the first-stage and maximum likelihood
estimates to form
" Trader submits an order | z t , Trader arrives .
Pr
(F5)
i
Let Z be the the state space for zt . Integrating over Z and using equations
(59) and (60) in the text,
1=
Pr (z t | Trader arrives) dzt
Z
=
Pr (z t | Trader arrives and submits and order) × ω(z t ) dzti .
Z
= Pr (Trader submits an order | Trader arrives)
Pr (z t | Trader arrives and submits and order)
×
dzt ,
Z Pr (Trader submits an order | z t , Trader arrives)
(F6)
Estimating the Gains from Trade
2803
where the last line follows from the definition of ω(zt ) in equation (60) and the
fact that
Pr (Trader submits an order | Trader arrives)
does not depend on zt . Equation (F6) implies that
Pr (Trader submits an order | Trader arrives)
−1
Pr (z t | Trader arrives and submits and order)
=
. (F7)
dzt
Z Pr (Trader submits an order | z t , Trader arrives)
Substituting in the empirical distribution for Pr (zt | Trader arrives and submits
an order) in the numerator and our estimate in equation (F5) in the denominator, we form
"
Pr(Trader
submits an order | Trader arrives)
−1
I 1
1
=
.
" Trader submits an order | z t , Trader arrives
I
Pr
i
i=1
(F8)
Using equations (F5) and (F8), we form the consistent estimate
ω̂(z ti ) =
"
Pr(Trader
submits an order | Trader arrives)
.
"
Pr Trader submits an order | z t , Trader arrives
(F9)
i
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